generalized fourier transforms and their applications a thesis submitted to the

GENERALIZED FOURIER TRANSFORMS AND THEIR

APPLICATIONS

A Thesis

Submitted to the Faculty

in partial fulfillment of the requirements for the

degree of

Doctor of Philosophy

in

Mathematics

by

Sarah Wolff

DARTMOUTH COLLEGE

Hanover, New Hampshire

May 2015

Examining Committee:

Daniel Rockmore, Chair

Thomas Shemanske

Sergi Elizalde

Richard Foote

F. Jon Kull, Ph.D.Dean of Graduate Studies

Copyright bySarah Wolff

2015

Abstract

This thesis centers around a generalization of the classical discrete Fourier transform.

We first present a general diagrammatic approach to the construction of efficient

algorithms for computing the Fourier transform of a function on a finite group or

semisimple algebra. By extending work which connects Bratteli diagrams to the

construction of Fast Fourier Transform algorithms [65], we make explicit use of the

path algebra connection to the construction of Gel’fand-Tsetlin bases and work in

the setting of general semisimple algebras and quivers. We relate this framework to

the construction of a configuration space derived from a Bratteli diagram. In this

setting the complexity of an algorithm for computing a Fourier transform reduces to

the calculation of the dimension of the associated configuration space.

We give explicit counting results to find the dimension of these configuration

spaces, and thus the complexity of the associated Fourier transform. Our methods

give improved upper bounds for the general linear groups over finite fields, the classical

Weyl groups, and homogeneous spaces of finite groups, while also recovering the best

known algorithms for the symmetric group and compact Lie groups. We extend these

results further to semisimple algebras, giving the first results for non-trivial upper

bounds for computing Fourier transforms on the Brauer and Birman-Murakami-Wenzl

(BMW) algebras.

ii

The extension of our algorithm to Fourier transforms on semisimple algebras is

motivated by emerging applications of such transforms. In particular, Fourier trans-

forms on the Iwahori-Hecke algebras have been used to analyze Metropolis-based

systematic scanning strategies for generating Coxeter group elements [25]. We con-

sider the Metropolis algorithm in the context of the Brauer and BMW monoids and

provide systematic scanning strategies for generating monoid elements. As the BMW

monoid consists of tangle diagrams, these scanning strategies can be rephrased as

random walks on links and tangles. We translate these walks into left multiplication

operators in the corresponding BMW algebra. Taking this algebraic perspective en-

ables the use of tools from representation theory to analyze the walks; in particular,

we develop a norm arising from a trace function on the BMW algebra to analyze the

time to stationarity of the walks.

iii

Acknowledgements

First and foremost, I’d like to thank my advisor, Dan Rockmore, for believing in

me long before I ever could. Thank you for your advice and guidance, for somehow

making time no matter how busy your schedule, and for your ability to spark a

month’s worth of ideas in thirty minutes (or less!).

A huge thank you as well to the members of my thesis committee: Sergi Elizalde,

Richard Foote, and Tom Shemanske, for your helpful comments, questions, and ideas

for future research directions. Thank you to Tom for being the person to run to

for any kind of math/life advice, despite never actually volunteering for this position.

Thank you to Sergi for not only being someone to look up to professionally but also for

showing me that it is possible to fit some soccer games into a successful mathematical

career. A million thanks as well to Arun Ram for inspiration, encouragement, and

helpful suggestions.

Thank you to my friends and teammates in Hanover for making the last five years

so much fun, and to my friends outside of Hanover for supporting me in my ‘quest

to understand paranomials.’ Thanks to Megan, Nathan, Lin, Emily, Jessie, Hannah,

and Ali. And to my Texan, for always being there, even when neither of us knew.

Above all else, thank you to my family, especially my mom and dad, for their love

and support. I love you and could not have done this without you.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

2 Fourier Transforms on Finite Groups and Semisimple Algebras 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Gel’fand-Tsetlin Bases and Bratteli diagrams . . . . . . . . . 14

2.2.2 Adapted Representations and Gel’fand-Tsetlin Bases . . . . . 22

2.2.3 The Separation of Variables Idea . . . . . . . . . . . . . . . . 24

2.3 The Separation of Variables Approach . . . . . . . . . . . . . . . . . 31

2.3.1 The Spaces Wi . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.2 The Bilinear Maps ∗ . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.3 The Quiver Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.4 The SOV Algorithm . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.5 General Morphism Counts . . . . . . . . . . . . . . . . . . . . 56

2.3.6 Morphisms into Locally Free Bratteli Diagrams . . . . . . . . 59

2.4 The Complexity of Fourier Transforms on Finite Groups . . . . . . . 65

v

2.4.1 The Weyl Groups Bn and Dn . . . . . . . . . . . . . . . . . . 65

2.4.2 The General Linear Group . . . . . . . . . . . . . . . . . . . . 73

2.4.3 Generalized Symmetric Group Case . . . . . . . . . . . . . . . 82

2.4.4 The Complexity of Fourier Transforms on Homogeneous Spaces 84

2.5 Extension to Semisimple Algebras . . . . . . . . . . . . . . . . . . . . 87

2.5.1 Background: The Brauer Algebra . . . . . . . . . . . . . . . . 88

2.5.2 Generalized SOV Approach . . . . . . . . . . . . . . . . . . . 91

2.5.3 The Brauer Algebra . . . . . . . . . . . . . . . . . . . . . . . 92

2.5.4 The BMW Algebra . . . . . . . . . . . . . . . . . . . . . . . . 96

2.5.5 General Result . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3 Random Walks on the Birman-Murakami-Wenzl Algebra 98

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2 Preliminaries: Probability Theory . . . . . . . . . . . . . . . . . . . . 103

3.2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.2.2 The Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . 106

3.2.3 Systematic Scans . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.3 Preliminaries: Semisimple Algebras . . . . . . . . . . . . . . . . . . . 109

3.3.1 Fourier Inversion and Plancherel . . . . . . . . . . . . . . . . . 109

3.3.2 The Brauer Algebra . . . . . . . . . . . . . . . . . . . . . . . 111

3.3.3 The BMW Algebra . . . . . . . . . . . . . . . . . . . . . . . . 113

3.4 The Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.5 Analysis of the Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Future Directions 137

vi

4.1 Further Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A Appendix 140

A.1 Gel’fand-Tsetlin Bases and Adapted Representations . . . . . . . . . 140

A.2 Restricted Product Lemmas . . . . . . . . . . . . . . . . . . . . . . . 143

A.3 Quiver Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

A.3.1 Smoothing Quivers . . . . . . . . . . . . . . . . . . . . . . . . 149

A.3.2 Properties of Locally Free Quivers . . . . . . . . . . . . . . . . 150

A.4 Combinatorial Lemmas for the Weyl Groups and Brauer Algebra . . . 154

A.4.1 The Weyl Group Bn . . . . . . . . . . . . . . . . . . . . . . . 154

A.4.2 The Weyl Group Dn . . . . . . . . . . . . . . . . . . . . . . . 160

A.4.3 The Brauer Algebra Brn . . . . . . . . . . . . . . . . . . . . . 166

A.5 Factoring Coset Representatives of GLn(Fq) . . . . . . . . . . . . . . 168

B Appendix 177

B.1 Example of Walk in BMW3 . . . . . . . . . . . . . . . . . . . . . . . 177

B.2 Symmetric Group Elements . . . . . . . . . . . . . . . . . . . . . . . 178

References 181

vii

Chapter 1

Introduction

This thesis unites discrete mathematics with representation theory and algebraic

combinatorics through the analysis and application of the construction of efficient

algorithms for the computation of a generalization of the classical discrete Fourier

transform (DFT) in the setting of finite groups and semisimple algebras.

Since Cooley and Tukey’s (re-)discovery of Gauss’ efficient algorithm for comput-

ing Fourier transforms, there have been far-reaching applications of the classical DFT

in many areas, including (but not limited to) digital signal processing, image process-

ing, spectral analysis, data compression, and astrophysics [2, 3, 11, 19, 31, 89, 91].

A generalized Fourier transform extends the concept behind the DFT to algebraic

structures wherein applications have been found in many domains, including VLSI

design, the design of filters, machine learning, conditional probability models, group

convolution algorithms, and crossover designs (see, e.g., [4, 6, 14, 80, 53, 55, 90, 97]).

In this thesis, we focus on generalized Fourier transforms; in particular, we develop

efficient algorithms to compute Fourier transforms on finite groups and semisimple

algebras, and also analyze applications to random walks on the Brauer and Birman-

1

Introduction

Murakami-Wenzl (BMW) algebras.

An algebraic framing of the DFT allows for its generalization to groups and

semisimple algebras. We view the DFT map that arises from expressing a function

f : Z/NZ→ C as a sum of complex exponentials:

f(j) =N−1∑k=0

f(k)ζk(j).

Here, Z/NZ = {0, 1, . . . , N − 1} is identified with the cyclic group of order N and

ζk : Z/NZ → C is defined by ζk(j) = 1√Ne2πijk/N . The coefficient f(k) is called the

kth Fourier coefficient of f.

The group algebra of Z/NZ, denoted C[Z/NZ], is the space of formal complex

linear combinations of group elements. By identifying complex-valued functions on

Z/NZ with elements of the group algebra

f ←→∑

j∈Z/NZ

f(j)j,

the DFT can be seen as a change of basis of C[Z/nZ] from the natural basis of

indicator functions {δk | k ∈ Z/NZ} to the orthonormal basis {ζk | k ∈ Z/NZ}:

∑j∈Z/nZ

f(j)jDFT−−−→

∑j∈Z/nZ

f(j)ζj

(for ζj =∑

k∈Z/nZ

ζj(k)k). In signal processing, f(j) is identified with a sampole of some

continuous function f : R → R and the DFT effects a Fourier representation of the

function f , thought of as a map from the time domain to the frequency domain [71].

2

Introduction

There is a natural inner product on C[Z/NZ] given by 〈f, g〉 =N−1∑j=0

f(j)g(j). The

orthonormality of the complex exponentials {ζk | k ∈ Z/NZ} under this inner product

allows for the determination of the kth Fourier coefficient as an inner product,

f(k) = 〈f, ζk〉 =N−1∑j=0

f(j)ζk(j).

Moreover, the functions {ζk} comprise a complete set of inequivalent irreducible

representations of Z/NZ. It is this beautiful result that underlies the generalization

of the DFT to finite groups and semisimple algebras: consider an arbitrary finite

group G and let f be a complex-valued function on G. Given a matrix representation

ρ of G, the (generalized) Fourier transform of f at ρ is given by the complex matrix

sum:

f(ρ) =∑x∈G

f(x)ρ(x).

The computation of the set of Fourier transforms of f at each representation

in a complete set R of inequivalent irreducible matrix representations of G is the

computation of a (generalized) Fourier transform of f at R. The problem of efficient

computation of generalized Fourier transforms is an area of active research (see e.g.

[6, 13, 15, 26, 54]). Some of the first results in this area are due to Willsky [94],

motivated by a search for efficient algorithms for filter design.

Let TG(R) denote the computational complexity of the Fourier transform on a

group G at a set of inequivalent irreducible representations R. This is basically the

least upper bound for the number of arithmetic operations needed to compute the

Fourier transform of an arbitrary complex-valued function f on G at R. We let C(G)

3

Introduction

denote the complexity of the group G, defined as

C(G) := minR{TG(R)}.

For N a “highly composite” number, Cooley and Tukey’s algorithm gives

C(Z/NZ) ≤ O(N log2N). More recently, Johnson and Frigo [51] and Lundy and

Van Buskirk [59] have independently shown that C(Z/NZ) ≤(

83

+ o(1))N log2N

for N = 2m. More generally, for A an abelian group of size N , it is known that

C(A) ≤ O(N log2N) [23], versus the naive bound, C(A) ≤ N2 that comes from

considering a direct computation.

Cooley and Tukey give a divide-and-conquer algorithm whose key step rewrites

the DFT on Z/NZ as a linear combination of DFTs on subgroups. This step applies

in the nonabelian group setting as well: if H is a subgroup of any group G, the

Fourier transform of a complex-valued function on G reduces to a linear combination

of Fourier transforms onH. This basic generalization of Cooley and Tukey’s algorithm

has been used in [65] to give efficient algorithms for generalized Fourier transforms and

has been extended to develop efficient algorithms for computing Fourier transforms

on finite inverse semigroups [61].

A more sophisticated approach that builds on the subgroup chain is given in

[63, 65], where a general algorithm is presented for computing Fourier transforms on

semisimple algebras as a sequence of bilinear maps written in terms of the path algebra

associated to a derived Bratteli diagram — a directed graph based on a subgroup chain

which depicts how representations factor when restricted to subgroups. In the path

algebra, basis vectors correspond to paths in the Bratteli diagram and matrix elements

(indexed by pairs of vectors) correspond to very specific kinds of subgraphs. Matrix

4

Introduction

element multiplication also has a diagrammatic formulation. Factorizations of the

associated Fourier transform correspond to decompositions of the Bratelli diagram

in terms of recurrent subgraphs. The complexity counts for an associated algorithm

then amount to counting the number of morphisms of subgraphs into the Bratteli

diagram.

The work in this thesis builds on these earlier ideas by using the formalism of

quivers [36]. We produce efficient algorithms for computing the Fourier transform of

functions on the general linear groups over finite fields, classical Weyl groups, and

their homogeneous spaces, improving on [63, 65]. We also provide new results for

general semisimple algebras including the Brauer and BMW algebras.

These new results are motivated in part by emerging applications of Fourier trans-

forms on semisimple algebras. For example, in [25] Diaconis and Ram consider the

problem of systematically generating elements of a finite Coxeter group, W . This

problem stated in terms of the group algebra C[W ] is equivalent to generating ele-

ments of the basis W of C[W ]. They show that a Metropolis-type formulation gives

rise to systematic scanning strategies — choices of an ordering of a succession of

Markov chains. This corresponds to the successive multiplication of group genera-

tors — which translates into an applications of left multiplication operators in the

Iwahori-Hecke algebra. The algebraic setting enables the use of Fourier analysis on

the Iwahori-Hecke algebra to analyze the time to stationarity of the corresponding

random walks. Their work gives a comparison between systematic scan algorithms

and random scan algorithms. This is of interest in the study of Ising models [10] and

Gibbs sampling for image processing [34].

We extend these Metropolis-type ideas to diagram algebras in the consideration

5

Introduction

of random walks on the monoid bases of the Brauer and BMW algebras. Elements of

these monoid bases have a natural interpretation as diagrams in which multiplication

becomes concatenation of diagrams. The Metropolis algorithm in the context of the

Brauer and BMW monoids gives rise to systematic scanning strategies for generating

basis elements via multiplication of generators. As the diagrams forming the BMW

monoid basis of the BMW algebra are tangles (see [40] or Chapter 3 for a description

of this basis), scanning strategies for generating BMW monoid elements have relations

to physics: random generation of links and tangles has been of use in [28, 60, 96]. As

in [25] the walks arising from the Metropolis algorithm can be reinterpreted as left

multiplication operators in the BMW algebra and studied via a new trace norm. This

allows us to use the tools of representation theory to analyze the time to stationarity.

The two parts of this thesis split naturally into two self-contained chapters. In

Chapter 2, we develop the Separation of Variables (SOV) approach for the efficient

computation of Fourier transforms on finite groups and semisimple algebras. We

end the chapter by using the algorithm to provide efficiency counts for the Brauer

and BMW algebras. In Chapter 3, we explore the application of Fourier transforms

on these algebras to random walks on the Brauer and BMW monoids. We close in

Chapter 4 with some thoughts about future directions of work.

6

Chapter 2

Fourier Transforms on Finite

Groups and Semisimple Algebras

2.1 Introduction

The (classical) Fast Fourier Transform (FFT) remains among the most important

family of algorithms in information processing [81]. In its simplest form, it effects the

efficient computation of the discrete Fourier transform (DFT), which up to normal-

ization and indexing is the computation, for each k ∈ Z/NZ, of the sums

f(k) =N−1∑j=0

f(j)e2πijk/N , (2.1)

as described in Chapter 1. While this calculation can be framed in a number of ways,

we take a representation theoretic point of view and cast the DFT as a change of basis

in C[Z/NZ], the complex group algebra of the cyclic group of order N (naturally

identified with Z/NZ), from the natural basis of indicator functions to a basis of

7

2.1 Introduction

irreducible matrix elements for Z/NZ. This perspective suggests a generalization of

the DFT to finite nonabelian groups G, that is, as the computation of a change of

basis in C[G] from the basis of indicator functions to a basis of irreducible matrix

elements, with concomitant questions of computational complexity (see e.g., [67]).

In the case of finite abelian groups, the various fast Fourier transform algorithms

can be pooled together in such a way that it is possible to say that the complexity

of the DFT on an abelian group of size N is bounded above by O(N log2N) [23].

Using different approaches, Lundy and Van Buskirk [59] and Johnson and Frigo [51]

independently determined complexity bounds of (83

+ o(1))N log2N for the case N =

2m. The deep and ongoing study of this problem has been motivated by a wide range

of applications in digital signal processing and beyond (see e.g. [2, 3, 11, 19, 31, 89,

91]).

The Cooley-Tukey algorithm is undoubtedly the most famous of the FFTs [17].

It is a divide-and-conquer algorithm, in fact, first recorded by Gauss in unpublished

work (see e.g. [47] for a brief history of the algorithm). The key step is to rewrite the

DFT on a cyclic group Z/NZ as a linear combination of DFTs on mZ/NZ < Z/NZ

(for N = mn). Iterating this step for a chain of subgroups of Z/NZ yields efficient

algorithms. In this chapter we continue a line of work that generalizes this approach

to nonabelian groups [64, 80, 65, 66], finding subgroup factorizations compatible with

bases of irreducible matrix elements that combine to provide the elementary ingredi-

ents of efficient DFT algorithms for arbitrary finite groups and more general algebraic

structures.

In particular, we build on the “Separation of Variables” (SOV) approach [65]

by integrating the approach detailed in [63] with the formalism of quivers. This

8

2.1 Introduction

gives a general algorithm for computing Fourier transforms on semisimple algebras

as a sequence of bilinear maps written in terms of the path algebra associated to a

derived Bratteli diagram — a directed graph based on a subgroup chain which depicts

how representations factor when restricted to subgroups. In the path algebra, basis

vectors correspond to paths in the Bratteli diagram, matrix elements correspond to

very specific kinds of subgraphs. Factorizations of the associated Fourier transform

correspond to decompositions of the Bratelli diagram in terms of recurrent subgraphs.

The complexity counts for an associated algorithm amount to counting the number

of morphisms of these subgraphs into the Bratteli diagram.

We use quivers to define spaces associated to morphisms of a graph with dimension

equal to the number of morphisms of the graph into the Bratteli diagram. The Fourier

transform is then realized as an accumulation of sequences of bilinear maps on these

“configuration spaces”. The associated complexity of such an algorithm is derived by

identifying each subgraph in a “gluing process” and computing the dimension of its

configuration space. As in [65], if our choice of factorization makes use of subgroup

and centralizer algebra structure, the configuration spaces of the subgraphs identified

in the gluing process have low dimension. With well chosen subgroup (subalgebra)

chains, we improve on upper bounds for the complexity of computing DFTs on the

general linear groups over finite fields, classical Weyl groups, and the homogeneous

spaces of these groups. Moreover, our results hold for arbitrary chains of semisimple

algebras rather than just chains of group algebras.

In Section 2.2 we outline the preliminaries needed for our results, defining the

Fourier transform and complexity and discussing the main ideas behind the SOV

approach. We then state our new complexity results for Fourier transforms on the

9

2.2 Background

general linear groups Gln(Fq), the Weyl groups Bn and Dn, and their homogeneous

spaces. In Section 2.3 we present the SOV approach in detail, rewriting an iterated

product in the path algebra as a sequence of bilinear maps on the newly defined con-

figuration spaces. The complexity of the algorithm is determined by the dimensions

of the configuration spaces and we generalize some results of Stanley on differential

posets [87, 88] to give explicit methods for finding these dimensions.

In Section 2.4 we give specific factorizations and counts to prove the specific group

complexity results and also recover previously known bounds for Sn [63] and compact

Lie groups [62]. Finally, we extend these ideas in Section 2.5 to general semisimple

algebras and work out complexity bounds for computing DFTs on the Brauer and

BMW algebras. The last of these provides material and motivation for Chapter 3.

2.2 Background

The usual discrete Fourier transform of a finite data sequence may be viewed as a

special case of Fourier transforms on finite groups, which in turn arise as a special

case of Fourier transforms on semisimple algebras. Here we review the basic concepts

and definitions. For more background on the representation theory of finite groups

we refer the reader to [86], while for semisimple algebras see [76]. We’ll work in the

general semisimple setting.

Definition 2.1. A matrix representation of a C-algebra A is an algebra homo-

morphism

ρ : A→Md(C),

where Md(C) denotes the complex algebra of d × d matrices with entries in C. We

10

2.2 Background

call d the dimension of ρ.

An algebra A is simple if A ∼= Mn(C) for some n ≥ 1 and semisimple if it

decomposes as a direct sum of simple algebras:

A ∼=⊕λ∈Λ

Mλ(C),

for a finite index set Λ.

Definition 2.2. Let A be a semisimple algebra, {ai}i∈I a basis for A and f =∑i∈I

f(ai)ai ∈ A.

(i) Let ρ be a matrix representation of A. Then the Fourier transform of f at

ρ, denoted f(ρ), is the matrix sum

f(ρ) =∑i∈I

f(ai)ρ(ai).

(ii) Let R be a set of matrix representations of A. Then the Fourier transform

of f on R is the direct sum of Fourier transforms of f at the representations

in R:

FR(f) =⊕ρ∈R

f(ρ) ∈⊕ρ∈R

Mdim ρ(C).

Example 2.1. Our main interest will be the case in which A = C[G], the complex

group algebra for a finite group G. The group algebra C[G] is the space of all formal

complex linear combinations of group elements under the product

(∑s∈G

f(s)s

)(∑t∈G

h(t)t

)=∑s,t∈G

f(s)h(t)st.

11

2.2 Background

Elements of C[G] are in one-to-one correspondence with complex-valued functions

on G, and the group algebra product corresponds to convolution of functions. The

Fourier transform of∑

s∈G f(s)s←→ f : G→ C at a matrix representation ρ of G is

f(ρ) =∑s∈G

f(s)ρ(s).

This is equivalent to the d2ρ individual Fourier transforms at the corresponding matrix

elements

f(ρij) =∑s∈G

f(s)ρij(s).

When we compute the Fourier transform for a complete set of inequivalent irreducible

representations R of G we refer to the calculation as the computation of a Fourier

transform on G (with respect to R).

Definition 2.3. Let G be a finite group, R a set of matrix representations of G.

(i) A straight-line program is a list of instructions for performing the operations

×,÷,+,− on inputs and precomputed values [9].

(ii) The arithmetic complexity of a Fourier transform on R, denoted TG(R),

is the minimum number of complex arithmetic operations needed to compute

the Fourier transform of f on R via a straight-line program for an arbitrary

complex-valued function f defined on G. Let C(G) denote the complexity of

the group G:

C(G) := minR{TG(R)},

where R varies over all complete sets of inequivalent irreducible representations

of G.

12

2.2 Background

(iii) The reduced complexity, denoted tG(R), is defined by

tG(R) =1

|G|TG(R).

Given a finite-dimensional semisimple algebra A, we have analogous definitions

for the arithmetic complexity, TA(R), and reduced complexity, tA(R) = 1dim(A)

TA(R).

As in [63] we will always define the arithmetic complexity of an algorithm to be

the maximum of the number of complex multiplications and the number of complex

additions.

A complete set R of inequivalent irreducible matrix representations of a group G

determines a basis for C[G] and a Fourier transform determines f through the Fourier

inversion formula.

Theorem 2.1 (Fourier inversion (see e.g. [24])). Let G be a group, f a complex-valued

function on G, and R a complete set of inequivalent irreducible matrix representations

of G. Then

f(s) =1

|G|∑ρ∈R

dimρ Trace(f(ρ)ρ(s−1).

Thus, the Fourier transform is a change of basis algorithm to a basis of irreducible

matrix elements. Reorganization using the obvious change of basis matrix gives a

naive complexity bound. Let ρ1, . . . , ρm be a complete set of inequivalent irreducible

matrix representations of a group G of dimensions d1, . . . , dm, respectively. A direct

computation of a Fourier transform would require at most |G|∑

i d2i = |G|2 arithmetic

operations. Rewriting, for a direct computation we have

C(G) ≤ TG(R) ≤ |G|2.

13

2.2 Background

Fast Fourier transforms (FFTs) are algorithms for computing Fourier trans-

forms that improve on this naive upper bound. A priori, the number of operations

needed to compute the Fourier transform may depend on the specific representations

used.

Example 2.2. The classical DFT and FFT. For G = Z/NZ, identified with the

cyclic group of order N , all irreducible representations are 1-dimensional, defined by

ζk(j) = e2πijk/N for k = 0, . . . , N − 1. The corresponding Fourier transform on Z/NZ

is the usual discrete Fourier transform. Cooley and Tukey’s algorithm showed that for

N ‘highly composite,’ i.e., N factors completely as a product of small prime numbers,

C(Z/NZ) ≤ O(N log2N) [17].

2.2.1 Gel’fand-Tsetlin Bases and Bratteli diagrams

The fundamental idea underlying the Cooley-Tukey FFT is a divide-and-conquer ap-

proach dependent on the use of a subgroup. In the best-known case of the Fourier

transform on Z/NZ, when N = nm the subgroup mZ/NZ ∼= Z/nZ enables a factor-

ization of the associated DFT matrix and related computation. This can be framed in

terms of a factorization of the path algebra of the Bratteli diagram of the associated

subgroup chain Z/NZ > mZ/NZ > {0}, an idea which extends to the computation of

the DFT for general group algebras and their associated irreducible matrix elements

[65].

Let A be a complex semisimple algebra with chain of subalgebras

A = An > An−1 > · · · > A1 > A0 = C,

14

2.2 Background

where for ease of notation we let Ai > Ai−1 denote that Ai−1 is a subalgebra of Ai.

We will see that for any such chain of subalgebras there is a natural isomorphism with

a chain of path algebras arising from the Bratteli diagram associated to the chain.

We will take advantage of a formulation in terms of graded quivers.

Definition 2.4. A quiver Q is a directed multigraph with vertex set V (Q) and edge

set E(Q). For an arrow, e ∈ E(Q) from vertex β to vertex α, we call α the head of

e and β the tail of e.

Example 2.3. The directed multigraph of Figure 2.1 is an example of a quiver.

Figure 2.1: A quiver

Let Q be a quiver. For each e ∈ E(Q), let e+ denote the head of e and e− the tail

of e.

Definition 2.5. A quiver Q is graded if there is a function gr : V (Q) → N such

that gr(e+) > gr(e−), for each e ∈ E(Q).

Example 2.4. Figure 2.2 is an example of a graded quiver. Each vertex v is labeled

by its grading, gr(v).

15

2.2 Background

0

1

1

2

2

Figure 2.2: A graded quiver

Definition 2.6. A Bratteli diagram is a finite graded quiver such that:

(i) there is a unique vertex with grading 0, called the root,

(ii) if v ∈ V (Q) is not the root then v is the head of at least one arrow,

(iii) if v ∈ V (Q) does not have grading of maximum value then v is the tail of at

least one arrow,

(iv) for each e ∈ E(Q), gr(e+) = 1 + gr(e−).

Example 2.5. Note that the quiver of Figure 2.2 is not a Bratteli diagram. However,

a slight modification produces the Bratteli diagram of Figure 2.3.

0

1

1

2

2

Figure 2.3: A Bratteli Diagram

Let A be a semisimple algebra with chain of subalgebras

16

2.2 Background

A = An > An−1 > · · · > A1 > A0 = C.

To associate a Bratteli diagram to this chain we follow the language of [77]. Let ρ be

an irreducible representation of Ai, i.e., an irreducible Ai-module. Upon restriction

to Ai−1, ρ ↓Ai−1decomposes as a direct sum of irreducible Ai−1-modules. For γ an

irreducible representation of Ai−1, let M(ρ, γ) denote the multiplicity of γ in ρ ↓Ai−1.

Definition 2.7. Given the chain of subalgebras,

A = An > An−1 > · · · > A1 > A0 = C,

the (associated) Bratteli diagram is described by

(i) The vertices of grading i are labeled by the (equivalence classes of) irreducible

representations of Ai;

(ii) A vertex labeled by an irreducible representation γ of Ai−1 is connected to a

vertex labeled by an irreducible representation ρ of Ai by M(ρ, γ) arrows.

Example 2.6. Let A = MN(C). The identity map 1N is an irreducible representation

of A with representation space CN . Similarly, the identity map 1 is an irreducible

representation of C with representation space C. Then M(CN ,C) = N and so the

Bratteli diagram for the chain MN(C) > C is the quiver of Figure 2.4, with N paths

between the two vertices.

17

2.2 Background

...N

MN(C) C>

0N

Figure 2.4

Example 2.7. Figure 2.5 shows two examples of Bratteli diagrams, with the gradings

listed at the top.

012

0

C

0

1

2

>C[2Z/6Z]

0

1

2

3

4

5

>C[Z/6Z]

0123

∅

> CC[S1]>C[S2]>C[S3]

Figure 2.5

On the left we see the Bratteli diagram for a chain of group algebras for the group

Z/6Z:

C[Z/6Z] > C[2Z/6Z] > C.

On the right we see the Bratteli diagram for a chain of group algebras for the sym-

metric group S3:

C[S3] > C[S2] > C[S1] > C.

Note that we distinguish C[S1] from C only so that vertices at level i, i > 0, cor-

respond to representations of C[Si]. For the group algebra C[Z/NZ], irreducible

representations are naturally indexed at each level by the integers 0, . . . , N − 1, while

18

2.2 Background

for C[Sn], the irreducible representations are indexed by partitions of n (as deter-

mined by Young in [95]; see [49] for an introduction to the representation theory of

Sn). For N = mn, mZ/NZ = {0,m, . . . , (n− 1)m} < Z/mnZ = Z/NZ, while Si will

be the subgroup of Sn fixing the elements {i+ 1, . . . , n}. This gives a natural way to

embed C[Z/nZ] as a subalgebra of C[Z/NZ] and C[Si] as a subalgebra of C[Sn].

Both Bratteli diagrams in Figure 2.5 are examples of multiplicity-free diagrams

in that there is at most one edge from any vertex of grading i to any vertex of grading

i+ 1.

Given a chain of subalgebras with Bratteli diagram B, there is a canonical chain

of algebras associated to B called the chain of path algebras.

Definition 2.8. Let B be a Bratteli diagram. The path algebra (at level i),

denoted C[Bi], is the C-vector space with basis given by ordered pairs of paths in B of

length i which start at the root of B and end at the same vertex at level i in B.

The path algebra C[Bi] is an algebra under the multiplication that linearly by the

distributive laws extends (P,Q) ∗ (P ′, Q′) = δQP ′(P,Q′):

∑(P,Q)

aPQ(P,Q) ∗∑

(P ′,Q′)

bP ′Q′(P′, Q′) =

∑(∑Q

aPQbQQ′

)(P,Q′).

Pictorially, the path algebra multiplication corresponds to Figure 2.6. The first

arrow represents gluing the pairs of paths along identical middle paths Q = P ′ and

the second arrow represents summation over all possible gluings.

19

2.2 Background

−→ −→Q

P

Q′

P ′ Q′

Q = P ′

P

Q′

P

Figure 2.6: Multiplication in the Path Algebra

Example 2.8. By indexing matrix entries by pairs of paths in the Bratteli diagram

for Mn(C) > C (Figure 2.4), the path algebra multiplication is matrix multiplication.

Example 2.9. For the Bratteli diagram B of Figure 2.7 associated to the chain

C[S3] > C[S2] > C[S1] > C, let P1, P2, P3, P4 be the paths from the root to level 3 in

B, labeled from top to bottom. Then C[B3] has basis

{(P1, P1), (P2, P2), (P2, P3), (P3, P2), (P3, P3), (P4, P4)}.

∅

> CC[S1]>C[S2]>C[S3]

P1

P2

P3

P4

Figure 2.7: Paths P1, P2, P3, P4, with label on the last arrow of the path.

Note that for a vertex v, labeled by a representation ρ, the dimension of ρ is given

by the number of paths from the root to v. Moreover, each path corresponds to a

subgroup-equivariant embedding of C into the representation space of ρ (for more

20

2.2 Background

details, see Appendix A.1).

Further, C[Bi] embeds into C[Bi+1] as a subalgebra by mapping any pair of paths

(P,Q) ∈ C[Bi] to the sum ∑e

(e ◦ P, e ◦Q),

over all arrows e such that the tail of e is the head of P (equivalently, of Q), and ◦

denotes concatenation of paths.

Example 2.10. Let B be the Bratteli diagram of Example 2.9 associated to the

chain C[S3] > C[S2] > C[S1] > C with paths P1, P2, P3, P4 from the root to level 3.

Then C[B2] has basis {(p1, p1), (p2, p2)} for p1 and p2 the paths of Figure 2.8 from the

root to level 2 in B. The basis element (p1, p1) sits inside C[B3] as (P1, P1) + (P2, P2)

and similarly p2 corresponds with (P3, P3) + (P4, P4). Together this determines an

embedding C[B2] ↪→ C[B3].

∅

> CC[S1]>C[S2]>C[S3]

p1

p2

Figure 2.8: Paths p1 and p2, with label on the last arrow of path.

For a Bratteli diagram B with highest grading n, consider the chain of path

algebras associated to B:

C[Bn] > C[Bn−1] > · · · > C[B1] > C[B0] = C.

21

2.2 Background

It is not too difficult to see that the Bratteli diagram associated to the chain of path

algebras is B.

For example, for C[BN ] > C[B0] = C the path algebra chain of Example 2.6, letting

pairs of paths index matrix entries yields a canonical representation C[BN ]→Mn(C),

and under this identification the Bratteli diagram for the chain is exactly B.

In fact:

Lemma 2.1. Any chain of semisimple algebras is uniquely determined up to isomor-

phism by its Bratteli diagram.

For further explanation see Appendix A and Section 2.3 of [39].

Remark 2.1. Quivers were first introduced by Gabriel in the study of modular rep-

resentation theory [36]. Bratteli diagrams were first introduced to classify inductive

limits of C∗-algebras [8]. After Elliot’s use of Bratteli diagrams in the classification

of AF-algebras [32], these ideas motivated a program to classify C∗-algebras in terms

of their K-theory [82]. In terms of the representation theory of semisimple algebras,

Bratteli diagrams have been of interest because those satisfying certain properties

correspond to algebras satisfying the Jones Basic Construction [45, 46]. Bratteli

diagrams have also been used to explicitly construct complete sets of irreducible rep-

resentations that are analogs of Young’s seminormal form in the symmetric group,

and to describe restriction relations of representations [20, 43, 44, 56].

2.2.2 Adapted Representations and Gel’fand-Tsetlin Bases

As in [63, 65] we use adapted sets of bases to compute Fourier transforms efficiently.

22

2.2 Background

Definition 2.9. Given a group G with subgroup H ≤ G, a complete set R of inequiv-

alent irreducible matrix representations of G is H-adapted if there exists a complete

set RH of inequivalent irreducible matrix representations of H such that for all ρ ∈ R,

ρ ↓H=⊕

γs, for (not neccessarily distinct) representations γs in RH . The set R is

adapted to the chain

G = Gn > Gn−1 > · · · > G0

if for each 1 ≤ i ≤ n there is a complete set Ri of inequivalent representations of

Gi such that Ri is Gi−1-adapted and Rn = R. A set of bases for the representation

spaces that give rise to adapted representations is an adapted basis.

The notion of an adapted basis coincides with that of a set, for each 1 ≤ i ≤ n, of

Gi-equivariant maps between the representation spaces of representations in Ri and

those in Ri+1. For further details, see Appendix A.1.

For the FFT results of the following sections we assume the ability to construct

adapted sets of representations. This requirement is not a limitation, as any set of rep-

resentations is equivalent to an adapted set of representations. One such construction

is outlined in [65].

The analogous concept in the path algebra associated to the group algebra chain

under the isomorphism of Lemma 2.1 is a system of Gel’fand-Tsetlin bases :

Definition 2.10. For a Bratteli diagram B associated to a chain of subalgebras of a

semisimple algebra A, a system of Gel’fand-Tsetlin bases for B consists of a

collection of bases for the representation spaces {Vα| α ∈ V (B)} of the representations

corresponding to α indexed by paths from the root to α, along with maps from the

23

2.2 Background

paths to the basis vectors; i.e., a set of basis vectors along with knowledge of the path

corresponding to each vector.

Example 2.11. Let B be the Bratteli diagram of Figure 2.5 associated to the chain

C[S3] > C[S2] > C[S1] > C. Then for the paths P1, P2, P3, P4 defined in Example

2.9, a basis {wP2 , wP3} for the two-dimensional representation space V is part of a

system of Gel’fand-Tsetlin bases for B. Note that the entries of the matrix of this

representation are indexed by pairs {(wPi , wPj) | i, j = 1, 2} and so correspond to

basis elements of the path algebra C[B3].

Systems of Gel’fand-Tsetlin bases were originally developed by Gel’fand and Tset-

lin to calculate the matrix coefficients of compact groups [37]. Clausen was the first

to apply them to the efficient computation of Fourier transforms on finite groups [13].

In Remark A.1 of Appendix A.1, we show that systems of Gel’fand-Tsetlin bases

for the chain of path algebras corresponding to a group algebra chain are equivalent

to adapted bases for the chain of subgroups.

2.2.3 The Separation of Variables Idea

Gel’fand-Tsetlin bases provide a means to better understand the isomorphism of

Lemma 2.1 between a chain of group algebras and the corresponding chain of path

algebras. To see this, first note that the Fourier transform of a function f on G

with respect to a complete set of inequivalent irreducible representations R of G is

an algebra isomorphism

C[G]FR−−−−→

⊕ρ∈R

Mdim(ρ)(C),

24

2.2 Background

where

f ←→∑s∈G

f(s)sFR−−−−→

⊕ρ∈R

f(ρ) =⊕ρ∈R

∑s∈G

f(s)ρ(s).

Now for a chain of subgroups G = Gn > Gn−1 > · · · > G0 = {e}, let B be the

Bratteli diagram of the chain of subalgebras C[Gn] > C[Gn−1] > · · · > C[G0] = C,

with corresponding chain of path algebras C[Bn] > C[Bn−1] > · · · > C[B1] > C.

Indexing the matrices ρ(s) by paths of length n in the Bratteli diagram corresponding

to a Gel’fand-Tsetlin basis for B, the isomorphism above becomes

∑s∈G


⊕ρ∈R

∑s∈G

f(s)ρ(s),

where ρ represents the matrix ρ written with respect to a Gel’fand-Tsetlin basis for

B.

Since Gel’fand-Tsetlin bases are indexed by paths in B and a basis for the path

algebra C[Bn] consists of pairs of paths, we identify the group algebra C[G] with its

realization in coordinates relative to the Gel’fand-Tsetlin basis, indexed by pairs of

paths of length n in B that share the same endpoint. Then

∑s∈G

f(s)s :=∑s∈G

f(s)∑

(P,Q)∈C[Bn]

[s]P,Q(P,Q)←→∑s∈G

f(s)s,

and so

∑s∈G

f(s)s←→∑s∈G


⊕ρ∈R

f(ρ) =⊕ρ∈R

∑s∈G

f(s)ρ(s).

In other words, as in [13, 63]:

Lemma 2.2. For C[G] = C[Gn] > C[Gn−1] > · · · > C[G1] > C[G0] = C a chain of

25

2.2 Background

subalgebras for C[G] with Bratteli diagram B, and R a complete set of inequivalent

irreducible representations of G adapted to the chain of subgroups Gn > Gn−1 > · · · >

G0, the computation of the Fourier transform of a function f on G with respect to R

is equivalent to computation of ∑s∈G

f(s)s

in the group algebra, relative to the Gel’fand-Tsetlin basis for B associated to R.

Identifying the group algebra with its realization in coordinates with respect to this

basis, this computation is equivalent to computation of

∑s∈G

f(s)s

in the path algebra C[Bn].

Example 2.12. Young’s orthogonal form gives an example of a complete set of ir-

reducible matrix representations for Sn adapted to the chain Sn > Sn−1 > · · · > S1.

Since restriction of representations from Sn to Sn−1 is multiplicity-free, the basis vec-

tors of a system of Gel’fand-Tsetlin bases for the irreducible representations relative

to this chain are determined up to scalar multiplies, and in the case of n = 3, the paths

are the paths P1, P2, P3, P4 of Example 2.9. In [63], Maslen gives an efficient algo-

rithm for computation of the Fourier transform of a function on Sn by considering the

computation of∑s∈Sn

f(s)s in the group algebra for Sn relative to this Gel’fand-Tsetlin

basis.

One motivation for Lemma 2.2 is that translation of matrix sums into path algebra

elements takes advantage of the direct sum structure of these matrices in that the

matrix elements corresponding to paths ending at different vertices in the Bratteli

26

2.2 Background

diagram (and thus zero) are ignored in the path algebra. For matrices with a more

structured form, e.g., those lying in subalgebras or centralizer algebras, the nonzero

entries are indexed by specific pairs of paths in the Bratteli diagram. Further, recall

that the definition of multiplication in the path algebra corresponds to gluing and

summing operations on quivers. Then to compute a product of elements in the path

algebra, we glue together subquivers corresponding to each element in the product

and find that the complexity estimates can be computed in terms of counts of the

number of such configurations in the Bratteli diagram. This was done in [63] for

the symmetric group case. Our translation to the path algebra and a reframing in

terms of quivers and configuration spaces enables an extension to arbitrary groups

and semisimple algebras.

Key to this is the introduction in Section 2.3 of a diagrammatic technique of gluing

and summing quivers to keep track of the nonzero entries.

The first steps of the SOV approach involve expressing a path algebra element as

a factorization over subsets of the Bratteli diagram in such a way as to disentangle the

dependencies in the sum f . To do so we first factor the Fourier transform through the

subalgebras C[Gi]. We illustrate with an example before stating the general result.

Example 2.13. To compute∑

s∈S3f(s)s, consider the set Y = {e, t1t2, t2} ⊆ S3, for

ti the simple transposition (i i+1). This is a complete set of left coset representatives

for S3/S2. In particular, S3 = {ee, et1, t1t2e, t1t2t1, t2e, t2t1}. The reason for keeping

the identity element e in the factorizations will become apparent. Then

27

2.2 Background

∑s∈S3

f(s)s = f(ee)ee+ f(et1)et1 + f(t1t2e)t1t2e+

f(t1t2t1)t1t2t1 + f(t2e)t2e+ f(t2t1)t2t1

=∑y∈Y

y∑s∈S2

f(ys)s.

Thus, the Fourier transform of f on C[S3] becomes a linear combination of Fourier

transforms of f on C[S2].

For G = Sn a set of coset representatives for Sn/Sn−1 is

Y = {e, t1 · · · tn−1, t2 · · · tn−1, . . . , tn−1}

and the same argument as above can be used in this general case.

We extend this idea to a general group G given a subgroup H and a set of coset

representatives Y for G/H.

Recall that TG(R) is defined to be the minimum number of operations to compute

the Fourier transform of a function f on G with respect to a set of representations R

of G. Recall also that

tG(R) :=1

|G|TG(R).

For H < G, let B be the Bratteli diagram of the group algebra chain C[G] > C[H] >

C, with corresponding path algebra chain C[BG] > C[BH ] > C. For Y a set of coset

representatives for G/H and Fy arbitrary elements of C[BH ], let

mG(R, Y,H) =1

|G|×

minimum number of operations required to compute∑y∈Y yFy in a system of Gel’fand-Tsetlin bases for B

28

2.2 Background

Lemma 2.3 is a restatement of Lemma 2.10 of [63] and Proposition 1 of [26].

Lemma 2.3. Let H be a subgroup of G, R a complete H-adapted set of inequivalent

irreducible matrix representations of G, and Y ⊆ G a set of coset representatives for

G/H. Let B be the Bratteli diagram of the group algebra chain C[G] > C[H] > C,

with corresponding path algebra chain C[BG] > C[BH ] > C. Then

tG(R) ≤ tH(RH) +mG(R, Y,H).

Proof. By Lemma 2.2, computation of the Fourier transform of a function f on G

at R is equivalent to computation of F :=∑

s∈G f(s)s in C[BG] by expressing group

algebra elements in coordinates relative to a Gel’fand-Tsetlin basis for B. Let H be

a subgroup of G and Y ⊆ G a set of coset representatives for G/H. Then

F :=∑s∈G

f(s)s =∑y∈Y

∑h∈H

f(yh)yh =∑y∈Y

y∑h∈H

f(yh)h

=∑y∈Y

yFy, (2.2)

where for each y ∈ Y,

Fy =∑h∈H

fy(h)h ∈ C[BH ]

with fy(h) := f(yh). Then to compute F , first compute Fy ∈ C[Bn−1] for all y ∈ Y

relative to a system of Gel’fand-Tsetlin bases for the chain C[BH ] > C corresponding

to RH , by means of a Fourier transform on H. This requires at most |G||H|TH(RH)

scalar operations. Next, express the elements Fy in coordinates relative to a system of

Gel’fand-Tsetlin bases for the path algebra chain C[BG] > C[BH ] > C corresponding

to R, which requires no operations. Finally, compute F using (2.2), which requires

29

2.2 Background

at most |G|mG(R, Y,H) operations. Thus,

TG(R) ≤ |G||H|

TH(RH) + |G|mG(R, Y,H),

and dividing by |G| proves the lemma.

By Lemma 2.3, to compute the Fourier transform of a complex function at a set

of H-adapted representations, we need to compute

FY =∑y∈Y

yFy,

for Y a set of coset representatives for G/H. Then to compute the Fourier transform

of a complex function at a set of representations R adapted to a chain G = Gn >

Gn−1 > · · · > G0 = e, let Yi be a set of coset representatives for Gi/Gi−1. Iteration

of Lemma 2.3 gives

tG(R) ≤ tG0(RG0) +n∑i=1

MGi(RGi , Y,Gi−1). (2.3)

In Section 2.3 we detail an approach for computing FYi . Applying this approach

yields the following theorems, proved in Section 2.4:

Theorem 2.2. For the Weyl group Bn and R a complete set of irreducible matrix

representations of Bn adapted to the subgroup chain Bn > Bn−1 > · · · > B0 = {e},

C(Bn) ≤ TBn(R) ≤ n(2n− 1)|Bn|.

Theorem 2.3. For the Weyl group Dn and R a complete set of irreducible matrix

30

2.3 The Separation of Variables Approach

representations of Dn adapted to the subgroup chain Dn > Dn−1 > · · · > D0 = {e},

C(Dn) ≤ TDn(R) ≤ n(13n− 11)

2|Dn|.

Theorem 2.4. For the matrix group Gln(Fq) and R a complete set of irreducible ma-

trix representations of Gln(Fq) adapted to the subgroup chain Gln(Fq) > Gln−1(Fq) >

· · · > {e}

C(Gln(Fq)) ≤ TGln(Fq)(R) = O(qn)|Gln(Fq)|.

We also consider the case of Fourier transforms on homogeneous spaces, producing

results such as:

Theorem 2.5. For the homogenous space Bn/Bn−k of the Weyl group Bn and R a

complete set of irreducible matrix representations of Bn adapted to the subgroup chain

Bn > Bn−1 > · · · > {e},

C(Bn/Bn−k) ≤ TBn/Bn−k(R) ≤ k(4n− 2k − 1)|Bn||Bn−k|

.


The SOV approach is comprised of three main steps:

1 Use Lemma 2.3 and Equation (2.3) to reduce computation of a Fourier transform

to computation of FY =∑

y∈Y yFy in the path algebra C[Bi], for Y a set of coset

representatives for Gi/Gi−1.

2 Rearrange FY to be a recursively structured summation.

31


3 Use diagrammatic techniques to keep track of nonzero entries and give com-

plexity counts.

We discussed Step 1 in Section 2.2.3. For Step 2, we develop a method to recur-

sively structure elements of the path algebra of the form∑

z∈Z⊆G

z. Let Z = {z | z ∈ Z}.

We begin with a preliminary algorithm to compute∑z∈Z

z.

Algorithm 2.1.

I. Choose m ∈ N and a subset X ⊆ (C[Bn])m = C[Bn] × · · · × C[Bn] such that

|X| = |Z| and

Z = {x1 · · ·xm|(x1, . . . , xm) ∈ X}.

Thus, X is a choice of factorization for each element of Z.

II. Let

Xi = {(xi+1, . . . , xm)|(x1, . . . , xm) ∈ X}, 0 ≤ i ≤ m,

Xm = ∅.

III. Define a sequence of functions Li : Xi → C[Bn] recursively by:

L1(x2, . . . , xm) =∑

(x1,x2,...,xm)∈X0

x1,

L2(x3, . . . , xm) =∑

(x2,x3,...,xm)∈X1

L1(x2, . . . , xm) ∗ x2.

Li(xi+1, . . . , xm) =∑

(xi,xi+1,...,xm)∈Xi−1

Li−1(xi, . . . , xm) ∗ xi.

Induction shows that Lm(∅) =∑z∈Z

z.

32


Example 2.14. Let G = S3 = {e, t1, t2, t1t2, t2t1, t1t2t1} and let B be the Bratteli

diagram corresponding to the chain C[S3] > C[S2] > C[S1] > C, with corresponding

path algebra chain C[B3] > C[B2] > C[B1] > C. Factor the elements of S3 as

S3 = {eee, t1ee, et2e, t1t2e, et2t1, t1t2t1}.

Let Z = S3. To compute∑z∈Z

z ∈ C[B3], let

X = {(e, e, e), (t1, e, e), (e, t2, e), (t1, t2, e), (e, t2, t1), (t1, t2, t1)}.

Then

X1 = {(e, e), (t2, e), (t2, t1)},

X2 = {(e), (t1)},

X3 = ∅,

and

L1(e, e) = L1(t2, e) = L1(t2, t1) = e+ t1,

L2(e) = L1(e, e)e+ L1(t1, e)t1 = ee+ t1e+ et2 + t1t2,

L2(t1) = L1(t2, t1)t2 = et2 + t1t2,

L3(∅) = L2(e)e+ L2(t1)t1 = eee+ t1ee+ et2e+ t1t2e+ et2t1 + t1t2t1 =∑z∈Z

z.

Algorithm 2.1 is the basis for Step 2 of the SOV approach. It provides a strat-

egy for recursively structuring the path algebra element∑

z∈Z z as a summation of

products, using factorizations of z.

We refine this approach further by recalling that the purpose of our translation

of matrix sums into path algebra elements was to take advantage of the fact that

33


matrices with a more structured form (e.g., those lying in subalgebras or centralizer

algebras) have nonzero entries indexed by specific paths in the Bratteli diagram. By

using factorizations of z that take advantage of this structure, we develop an efficient

algorithm for the computation. To do so, for a factorization z = x1x2 · · ·xm, we keep

track of the space in which each element xi lies. Further, we translate multiplication

into a bilinear map on these vector spaces, the complexity of which is easily computed.

To give the general idea, recall that the definition of multiplication in the path

algebra (Section 2.2.1) corresponds to the natural gluing and summing operations of

Figure 2.9, where the first arrow represents gluing the pairs of paths along identical

middle paths Q = P ′ and the second arrow represents summation over all possible

gluings.

−→ −→Q

P

Q′

P ′ Q′

Q = P ′

P

Q′

P

Figure 2.9: Multiplication in the Path Algebra

Then to compute a product of elements in the path algebra, we form a quiver Q by

gluing together quivers that correspond to each element in the product. Summing over

all possible gluings amounts to counting the occurrences of Q in the corresponding

Bratteli diagram B. We do so by defining the space of maps from Q into B, as the

dimension of this space is the number of occurrences of Q in B.

Example 2.15. Let G = S4 and consider the subalgebra chain C[S4] > C[S3] >

34


C[S2] > C[S1] > C[S0] = C with Bratteli diagram B and corresponding path algebra

chain C[B4] > C[B3] > C[B2] > C[B1] > C[B0]. Let Y = {e, s1s2s3, s2s3, s3}, a set of

coset representatives for S4/S3. Then by Lemma 2.2.3 and an extension of Example

2.13, to compute∑

s∈S4f(s)s we need to compute

∑y∈Y

yFy = eFe + t1t2t3Ft1t2t3 + t2t3Ft2t3 + t3Ft3 .

For the purposes of this example, consider the single term t1t2t3Ft1t2t3 .

Note that ti ∈ C[Si+1]∩CentralizerC[Si−1]. Identifing the group algebra with the

path algebra by expressing ti in coordinates, ti :=∑

(P,Q)[ti]PQ(P,Q), an application

of Schur’s lemma and standard facts about Gel’fand-Tsetlin bases show that [ti]P,Q is

0 unless P and Q are paths in B that agree from level 4 to level i+ 1, and from level

i − 1 to level 0. Thus, ti ∈ C[Bi+1] ∩ CentralizerC[Bi−1], isomorphic to the space of

maps from the quiver Qi of Figure 2.10 into B. For further details, see Lemma 2.4.

Qi

04 i− 1i+ 1

Figure 2.10

Moreover, multiplication in the path algebra corresponds to gluing paths, and so

the product t1t2t3Ft1t2t3 lies in a space isomorphic to the space of maps from the

quiver Q of Figure 2.11 into B.

35


Q

t1t2t3

Ft1t2t3

01

1

2

23

3

4

Figure 2.11

However, determining the maps from Q into B (i.e., the number of occurrences of

Q as a subquiver of B) can be complicated. The key is to instead use the recursive

sum strategy of Algorithm 2.1 in order to consider each multiplication individually.

We rewrite the product as individual gluing operations

(((Ft1t2t3 ∗ t1) ∗ t2) ∗ t3),

where w1 ∗w2 represents gluing together the subquivers of Q corresponding to w1 and

w2. Note that gluing Ft1t2t3 to t1 corresponds to the subquiver Q1 of Q (see Figure

2.12), while gluing (Ft1t2t4 ∗ t1) to t2 corresponds to the subquiver Q2 of Q, and gluing

((Ft1t2t4 ∗ t1) ∗ t2) to t3 corresponds to the subquiver Q3.

The complexity of the algorithm will result from counting the number of occur-

rences of the subquivers Qi in the Bratteli diagram B.

Qi

ti 0

3 i− 1i

ii+ 1

Figure 2.12

36


In general, to compute∑

z∈Z z, we proceed as in Example 2.15 by factoring each

z = x1 · · ·xm, determining the corresponding spaces Wi such that xi ∈ Wi, and

defining bilinear maps ∗ corresponding to gluing.

Algorithm 2.2.

I. Choose m ∈ N and a subset X ⊆ (C[Bn])m = C[Bn] × · · ·C[Bn] such that

|X| = |Z| and

Z = {x1 · · ·xm|(x1, . . . , xm) ∈ X}.

Thus, X can be thought of as a choice of factorization into at most m elements

of each element of Z .

II. For σ ∈ Sm, let wi = xσ(i). For 1 ≤ i ≤ m let Wi be a vector space containing

wi and for 2 ≤ j ≤ m let ∗j be a bilinear map such that

x1 · · ·xm = (((w1 ∗2 w2) ∗3 w3) · · · ∗m wm).

III. Let

Xi = {(wi+1, . . . , wm) | (x1, . . . , xm) ∈ X} ⊆ Wi+1 × · · · ×Wm, 0 ≤ i ≤ m,

Xm = ∅.

IV. Define a sequence of functions Li recursively by:

L1(w2, . . . , wm) =∑

(w1,w2,...,wm)∈X0

w1,

Li(wi+1, . . . , wm) =∑

(wi,wi+1,...,wm)∈Xi−1

(Li−1(wi, . . . , wm) ∗i wi).

37


Theorem 2.6. For Li as defined above,

Lm := Lm(∅) =∑

(w1,...,wm)∈X0

(((w1 ∗2 w2) ∗3 w3) · · · ∗m wm) =∑z∈Z

z

Proof. First note that

∑z∈Z

z =∑

(x1,...,xm)∈X

x1 · · ·xm =∑

(w1,...,wm)∈X0

(((w1 ∗2 w2) ∗3 w3) · · · ∗m wm).

Then induction gives that

∑(w1,...,wm)∈X0

(((w1 ∗2 w2) ∗3 w3) · · · ∗m wm)

=∑

(wi,...,wm)∈Xi−1

(((Li−1(wi, . . . , wm) ∗i wi) ∗i+1 wi+1) · · · ∗m wm),

so for i = m we have the theorem statement.

Note 2.1. When σ = e, i.e. wi = xi, and the bilinear map is multiplication, i.e.

(wi−1 ∗i wi) = wi−1wi, Theorem 2.6 is exactly the conclusion of Algorithm 2.1, illus-

trated by Example 2.14. The bilinear maps we define are essentially multiplication

of matrix entries, but will take advantage of sparse matrix structure.

In the remaining parts of Section 2.3 we flesh out Algorithm 2.2 by constructing the

required spaces Wi and bilinear maps ∗j. As elements of subalgebras and centralizer

algebras have nonzero entries indexed by specific paths in the Bratteli diagram (see

Example 2.15), these spaces will be intersections of such algebras.

38


2.3.1 The Spaces Wi

Let B be a Bratteli diagram with highest grading at least n and for x1, . . . , xm ∈ C[Bn],

consider the product x1 · · ·xm.

Definition 2.11. For a path algebra product x1 · · · xm, Let i+ denote the smallest

integer such that xi ∈ C[Bi+ ] and let i− denote the largest integer less than or equal

to i+ such that xi ∈ Centralizer(C[Bi− ]). Then for 1 ≤ i ≤ m define

Wi := C[Bi+ ] ∩ Centralizer(C[Bi− ]).

Note that xi ∈ Wi and the product x1 · · ·xm is a multilinear map

W1 × · · · ×Wm → C[Bn].

Example 2.16. Let B be the Bratteli diagram for the chain of symmetric group

algebras C[S4] > C[S3] > C[S2] > C[S1] > C and let C[B4] > C[B3] > C[B2] >

C[B1] > C be the corresponding chain of path algebras.

Consider the product x1x2x3x4 = t1t2t3Ft1t2t3 of Example 2.15, where ti is the

transposition (i i + 1) and Ft1t2t3 ∈ C[B3]. Since ti ∈ C[Bi+1] ∩ Centralizer(C[Bi−1]),

we have that

1+ = 2, 1− = 0,

2+ = 3, 2− = 1,

3+ = 4, 3− = 2.

Finally, we know that x4 = Ft1t2t3 is an arbitrary element of C[B3], implying that

4+ = 3 and 4− = 0.

We will show (Lemma 2.4 below), that each space Wi is isomorphic to the config-

39


uration space of a specific quiver.

For an arrow e of a quiver Q, let e+ ∈ V (Q) denote the head of e and e− the tail.

Definition 2.12. For graded quivers Q and B, a morphism φ : Q → B is a

mapping from arrows in Q to paths in B, along with a grading-preserving mapping

between vertices so that φ(e+) = φ(e)+ and φ(e−) = φ(e)− for all arrows e ∈ E(Q).

Example 2.17. For Q, B as in Figure 2.13, let φ : Q→ B send the arrow e1 to the

path f3 ◦ f2 ◦ f1.

Q B3 0 3 2 1 0

e1 f1f2f3

Figure 2.13

For two graded quivers Q and B, let Hom(Q;B) denote the set of morphisms

from Q to B. For Q,R, and B graded quivers such that Q is a subquiver of R, let

Hom(Q ↑ R;B) denote the set of morphisms from Q to B that extend to R.

Definition 2.13. The configuration space associated to Q and R relative to B,

denoted A(Q ↑ R;B), is the space of finitely supported formal C-linear combinations

of morphisms in Hom(Q ↑ R;B).

Note 2.2.

1. When Q = R, we simplify notation by writing A(Q;B).

2. If Q is a finite subquiver of R and B is locally finite, i.e. each vertex has finitely

many neighbors, then # Hom(Q ↑ R;B) = dimA(Q ↑ R;B).

40


Certain configuration spaces are naturally isomorphic to the spacesWi of definition

2.11:

Lemma 2.4. Let {Ai} be a chain of subalgebras of a semisimple algebra A with

corresponding Bratteli diagram B of highest grading at least n. Consider the quivers

Qn0 and Qnji of Figure 2.14, along with the subquiver Qji of Qn

ji consisting of the two

vertices at level i and level j, along with the two paths from level i to level j:

n 0

Qn0

n 0j i

Qji ↪→ Qnji

Figure 2.14

Then as vector spaces,

(i) A(Qn0;B) ∼= C[Bn] ∼= An,

(ii) A(Qji;B) = A(Qji ↑ Qnji;B) ∼= C[Bj]∩Centralizer(C[Bi]) ∼= Aj∩Centralizer(Ai).

Proof. (i) follows from the proof of (ii). Alternatively, note that morphisms φ : Qn0 →

B correspond to pairs of paths of length n in B starting at the root, 0, and ending at

the same vertex, i.e., basis elements of the path algebra C[Bn].

To prove (ii), note that morphisms φ : Qji → B correspond to pairs of paths in B

starting at the same vertex at level i and ending at the same vertex at level j.

Let wji ∈ Wji := Aj ∩ Centralizer(Ai). Recall that a pair of paths in B,

P = pn ← pn−1 ← · · · ← p1 ← 0,

41


Q = qn ← qn−1 ← · · · ← q1 ← 0,

with pn = qn determines an element of the Gel’fand-Tsetlin basis for C[Bn] corre-

sponding to the path algebra chain. Represent wji in its coordinates relative to the

Gel’fand-Tsetlin basis by [wji]PQ. Using standard facts about Gel’fand-Tsetlin bases

(see eg. [63, Lemma 4.1], [39, Proposition 2.3.12]),

[wji]PQ = δpn−1qn−1 · · · δpjqjP jiwjiδpiqi · · · δp1q1 ,

for P jiwji

a complex function on the vertices pj, . . . pi, qj, . . . , qi in B. Thus, under the

map

wji → P jiwji,

the space Wji is isomorphic to the space of complex functions on vertices pj, . . . pi,

qj, . . . , qi of B that form paths in B with pj = qj and pi = qi.

By Lemma 2.4, for the spaces Wi := C[Bi+ ] ∩ Centralizer(C[Bi− ]) defined above,

Wi∼= A(Qi+i− ;B).

2.3.2 The Bilinear Maps ∗

To understand multiplication of elements wi ∈ Wi, we first determine (Theorem 2.7)

that the bilinear map on A(Qi+i− ;B) corresponding to path algebra multiplication is

ι∗ ◦ φ∗ ◦⊗

, for ι and φ natural inclusions, and ι∗, φ∗, and⊗

defined below.

For B a graded quiver, ι : Q1 → Q2 an inclusion of graded quivers, and µ ∈

42


Hom(Q2;B), let µ ↓Q1 denote µ ◦ ι ∈ Hom(Q1;B).

Definition 2.14. For B a graded quiver and ι : Q1 → Q2 an inclusion of graded

quivers, define ι∗ : A(Q2;B) → A(Q1;B) by linearly extending the map that sends

µ ∈ Hom(Q2;B) to µ ↓Q1.

For Q1 a subquiver of R1, Q2 a subquiver of R2, and ι : Q1 → Q2 an inclusion

that extends to an inclusion from R1 to R2, analogously define ι∗ : A(Q2 ↑ R2;B)→

A(Q1 ↑ R1;B).

Definition 2.15. For B a locally finite quiver, Q2 a finite quiver, and ι : Q1 → Q2

an inclusion of graded quivers, define ι∗ : A(Q1;B)→ A(Q2;B) by linearly extending

the map that sends τ ∈ Hom(Q1;B) to∑

µ↓Q1=τ

µ.

As above, this definition may be extended to define ι∗ : A(Q1 ↑ R1;B)→ A(Q2 ↑

R2;B).

Example 2.18. Consider the quivers Q1, Q2, B of Figure 2.15 and the morphism φ :

Q1 → Q2 that sends q1 to p1. Note that Hom(Q1;B) = {G1, G2}, where G1(q1) = s1

and G2(q1) = s2. Further, Hom(Q2;B) = {H1, H2, H3, H4}, for

H1(p1) = s1, H1(p2) = s3,

H2(p1) = s1, H2(p2) = s4,

H3(p1) = s2, H3(p2) = s3,

H4(p1) = s2, H4(p2) = s4.

43


Q1 Q2 B

03 03

2

0

2

3

q1 p1

p2

s1

s2

s4

s3

Figure 2.15

Then for f =∑Hi

f |HiHi ∈ A(Q2;B),

φ∗(f) = f |H1G1 + f |H2G1 + f |H3G2 + f |H4G2.

Similarly, for g =∑Gi

f |GiGi ∈ A(Q1;B),

φ∗(g) = g|G1H1 + g|G1H2 + g|G2H3 + g|G2H4.

Denote the disjoint union of two directed graded graphs (quivers) Q1 and Q2

by Q1 t Q2. Then Q1 t Q2 is the quiver with vertex set V (Q1) t V (Q2), arrows

E(Q1) t E(Q2), and grading gr1 t gr2, where for v ∈ Q1 tQ2,

gr1 t gr2(v) =

gr1(v) if v ∈ Q1,

gr2(v) if v ∈ Q2.

Let ιj denote the natural inclusions Qj ↪→ Q1 tQ2, for j = 1, 2.

Definition 2.16. For R1 and R2 graded quivers with subquivers Q1 and Q2, respec-

tively, define

⊗: A(Q1 ↑ R1;B)× A(Q2 ↑ R2;B)→ A(Q1 tQ2 ↑ R1 tR2;B),

44


as follows: for (f, g) ∈ A(Q1 ↑ R1;B)× A(Q2 ↑ R2;B),

⊗(f, g) =

∑η∈Hom(Q1tQ2↑R1tR2;B)

f |η◦ι1g|η◦ι2η.

It is easily checked that⊗

is bilinear.

Recall the pictorial representation (Figure 2.6) of multiplication in the path al-

gebra. Theorem 2.7 translates this multiplication into a composition of maps on

configuration spaces, using the surjection φ := ι1 t ι2 and inclusion ι := ι4 of Figure

2.16, where 4 denotes the symmetric difference, defined below.

Qn0 tQn0 R = Qn0 ∪Qn0 Qn0 = Qn04Qn0

φ−→ ι←↩

0n

0n

n 0 0n

q

p

q′

p′ q′

q = p′

p

q′

p

Figure 2.16

Definition 2.17. For a graded quiver R with subquivers Q1, Q2, the symmetric

difference of Q1 and Q2 is

Q14Q2 = (Q1 \Q2) ∪ (Q2 \Q1) ∪ {isolated vertices of Q1 and Q2},

where (Qi \Qj) := Qi \ (Qi ∩Qj), i.e., the smallest subquiver of Qi that contains all

the vertices and arrows of Qi not in Qi ∩Qj.

45


Theorem 2.7. For B a Bratteli diagram of highest grading at least n, φ = ι1 t ι2,

and ι = ι4 as in Figure 2.16, the bilinear map

ι∗ ◦ φ∗ ◦⊗

: A(Qn0;B)× A(Qn0;B)→ A(Qn0;B)

corresponds to the algebra product on C[Bn] under the isomorphism A(Qn0;B) ∼= C[Bn]

of Lemma 2.4.

Proof. As the proof is little more than applying the definitions of ι∗ and φ∗, we have

deferred it to Appendix A.2.

Note 2.3. By Lemma 2.4,

C[Bi] ∩ CentralizerC[Bj] ∼= A(Qji;B)

and so by Theorem 2.7, for

x ∈ C[Bj] ∩ CentralizerC[Bi], y ∈ C[Bl] ∩ CentralizerC[Bk],

the product xy corresponds to the bilinear product

ι∗ ◦ φ∗ ◦⊗

: A(Qji;B)× A(Qlk;B)→ A(Qji4Qlk;B).

As an example see Figure 2.17.

46


Qnji tQn

lk Qnji ∪Qn

lk Qnji4Qn

lk

ι1 t ι2−→ι4←↩

0

0

n

n

0n n 0

i

i i

j

j j

l

l l

k

k k

Figure 2.17: i ≤ k ≤ l ≤ j

A product of two elements in the path algebra thus corresponds to bilinear maps

arising from natural injections and surjections on the configuration spaces of the

corresponding quivers.

Definition 2.18. For Q1 and Q2 graded quivers, ι4 the natural injection of Q14Q2

into Q1 ∪ Q2 and ι1 t ι2 the natural surjection of Q1 t Q2 into Q1 ∪ Q2, define the

restricted product ∗ : A(Q1;B) × A(Q2;B) −→ A(Q1 4 Q2;B) as follows: for

fi ∈ A(Qi;B),

∗(f1, f2) = (ι4)∗ ◦ (ι1 t ι2)∗ ◦⊗

(f1, f2).

We may think of the restricted product as gluing Q1 to Q2, embedding this into

Q1 ∪Q2, and deleting arrows in Q1 ∩Q2; hence, ‘restricting’ the product.

In Appendix A.2 we prove that the restricted product is associative and commu-

tative. Further, the complexity is easily determined:

Lemma 2.5. For B a locally finite graded quiver, R a graded quiver with finite

subquivers Q1 and Q2, and fi ∈ A(Qi;B), the restricted product f1 ∗ f2 requires at

most

# Hom((Q1 ∪Q2) ↑ R;B)

47


scalar multiplications and

# Hom((Q1 ∪Q2) ↑ R;B)−# Hom((Q14Q2) ↑ R;B)

scalar additions.

Proof. See appendix A.2 for proof.

By Theorem 2.7, for

xj ∈ Wj = C[Bj+ ] ∩ Centralizer(C[Bj− ]) ∼= A(Qj+j− ;B),

the product xixi+1 corresponds to a restricted product

∗ : A(Qi+i− ;B)× A(Q(i+1)+(i+1)− ;B)→ A(Qi+i− 4Q(i+1)+(i+1)− ;B),

where we identify edges of Qi+i− with edges of Q(i+1)+(i+1)− as in Figure 2.16. Then

multiplying this product by xi+2 corresponds to a restricted product where we identify

edges of Qi+i− 4Q(i+1)+(i+1)− with edges of Q(i+2)+(i+2)− .

Then to compute x1 · · · xm, we must make these identifications for all Qi+i− . To be

explicit, we do so all at once by associating a quiver Q. Each Qi+i−4Q(i+1)+(i+1)− is

a subquiver of Q so to compute the product x1 · · ·xm we consider restricted products

of elements in A(Q;B).

2.3.3 The Quiver Q

To construct Q, for each i, 1 ≤ i ≤ m, consider the quiver Qni+i− with arrows Li and

Ri as in Figure 2.18.

48


Qi+i− ↪→ Qni+i−

0n i−i+

Ri

Li

Figure 2.18

First, ‘unsmooth’ each quiver by adding in vertices that allow Ri to be glued to

Li+1: for each 1 ≤ i ≤ m, add vertices to Li and Ri at levels j+ and j− for all

1 ≤ j ≤ m. Note that it is not always necessary to add in all of these vertices to be

able to glue Ri to Li+1 (see Example 2.19).

Let Q be the quiver formed by gluing Ri to Li+1, 1 ≤ i < m. Note that each

quiver Qni+i− and Qi+i− injects naturally into Q, via the inclusion map, ι. Let

Qi := ι(Qni+i−), Fi := ι(Qi+i−), F := ∪mi=1Fi,

Li := ι(Li), for 1 ≤ i ≤ m, Lm+1 = ι(Rm).

Example 2.19. Suppose 1+ = 7, 1− = 4, 2+ = 3, 2− = 1, 3+ = 5, 3− = 2. Then we

have the quivers Qni+i− of Figure 2.19. We unsmooth Qn

74 by adding vertices at level

5, 3, and 1 to R1; we unsmooth Qn31 by adding a vertex at level 2 to R2 and vertices

at level 4, 5, and 7 to L2; we unsmooth Qn52 by adding vertices at level 7, 4, 3, and 1

to L3.

We then glue together Qn74 and Qn

31 by gluing R1 to L2 at the vertices at level

1, 3, 4 and 7. We then glue this quiver to Qn52 by gluing R2 to L3 along the vertices

at levels 1, 2, 3, 5. This gluing process creates the quivers Q and F , with subgraphs

Qi, and Fi as in Figure 2.20.

49


Qn74

Qn31

Qn52

Q

Q74Q31

Q52

F

Q74Q31

Q52

47

1

0n

3

2

0n

5 0n

01

2

34

5

7n

1

2

34

5

7

R1

L1

R2

L2

R3

L3

Figure 2.19

Q1

Q74

Q2

Q31

Q3

Q52

F1

Q74

F2

Q31

F3

Q52

0134

5

7n

01

2

34

5

7n

01

2

34

5

7n

4

5

7

1

2

3

2

34

5

Figure 2.20

Example 2.20. Consider the product t1t2t3Ft1t2t3 of Example 2.15 and recall from

Example 2.16 that

1+ = 2, 1− = 0,

2+ = 3, 2− = 1,

3+ = 4, 3− = 2,

4+ = 3, 4− = 0.

50


Then gluing together the quivers Q420, Q4

31, Q442, and Q4

30 (i.e. the quivers Q1, Q2,

Q3, QF of Example 2.15) yields the quiver Q of Example 2.15.

Recall from Lemma 2.4 that

A(Qi+i− ;B) ∼= C[Bi+ ] ∩ Centralizer(C[Bi− ]) = Wi.

Let ζi : Wi → A(Qi+i− ;B) be the isomorphism of Lemma 2.4. By Theorem 2.7,

for wi ∈ Wi, the algebra product w1 · · ·wm corresponds to the restricted product

ζ1(w1) ∗ · · · ∗ ζm(wm). To be explicit, we have a map

ζ−1 : A(Q1+1− 4 · · · 4Qm+m− ;B)→ C[Bn]

such that

ζ−1(ζ1(w1) ∗ · · · ∗ ζm(wm)) = w1 · · ·wm.

After smoothing Fi, Corollary A.2 in Appendix A.3.1 gives an isomorphism

χi : A(Qi+i− ;B)→ A(Fi;B),

which in turn allows for definition of a map:

χ−1 : A(F14 · · · 4 Fm;B)→ A(Q1+1− 4 · · · 4Qm+m− ;B)

such that for fi ∈ A(Qi+i− ;B),

χ−1(χ1(f1) ∗ · · · ∗ χm(fm)) = f1 ∗ · · · ∗ fm.

51


Thus, multiplication in the path algebra becomes a restricted product on config-

uration spaces corresponding to subquivers of Q.

Theorem 2.8. For wi ∈ Wi = C[Bi+ ] ∩ Centralizer(C[Bi− ]), 1 ≤ i ≤ m, and quivers

Q,Qi, F, Fi and maps χ−1, χi, ζ−1, ζi as defined above, let

θ = ζ−1 ◦ χ−1 : A(F14 · · · 4 Fm;B)→ C[Bn],

and let

ξi = χi ◦ ζi : Wi → A(Fi;B).

Then

w1 · · ·wm = θ(ξ1(w1) ∗ · · · ∗ ξm(wm)).

2.3.4 The SOV Algorithm

To apply Theorem 2.6 we define vector spaces Wi and Vi along with bilinear maps

∗j : Vj−1 ×Wj → Vj satisfying the conditions of Algorithm 2.2.

For σ ∈ Sm and 1 ≤ i ≤ m, let Rσi := Fσ(1)4 · · · 4 Fσ(i) ⊆ Q. Let

Wi := Wσ(i) = C[Bσ(i)+ ] ∩ Centralizer(C[Bσ(i)− ]),

V1 = W1, Vi = A(Rσi ;B), for 2 ≤ i < m, and Vm = C[Bn].

52


Define ∗j : Vj−1 ×Wj → Vj by

∗2(v1, w2) = v1 ∗2 w2 := (ξσ(1)v1) ∗ (ξσ(2)w2),

∗j(vj−1, wj) = vj−1 ∗j wj := vj−1 ∗ (ξσ(j)wj), for 2 < j < m,

∗m(vm−1, wm) = vm−1 ∗m wm := θ(vm−1 ∗ (ξσ(m)wm)).

Theorem 2.9. Let m,n ∈ Z+ and for 1 ≤ i ≤ m let i+, i− ∈ N such that i+, i− ≤ n.

For xi ∈ C[Bi+ ] ∩ Centralizer(C[Bi− ]), σ ∈ Sm, and wi := xσ(i) ∈ Wi,

1. x1 · · ·xm = (((w1 ∗2 w2) ∗3 w3) · · · ∗m wm)

2. This may be computed in at most

m∑i=2

dimA(Rσi−1 ∪ Fσ(i) ↑ Q)

scalar multiplications, and

m∑i=2

(dimA(Rσi−1 ∪ Fσ(i) ↑ Q)− dimA(Rσ

i ↑ Q))

scalar additions.

Proof. To prove Part 1, it follows by definition of ∗i and induction that for 2 ≤ i < m,

(((w1 ∗2 w2) ∗3 w3) · · · ∗i wi) = ξσ(1)(w1) ∗ · · · ∗ ξσ(i)(wi).

53


Then by commutativity of ∗ and Theorem 2.8,

(((w1 ∗2 w2) ∗3 w3) · · · ∗m wm) = ∗m(ξσ(1)(w1) ∗ · · · ∗ ξσ(m−1)(wm−1), wm)

= θ(ξσ(1)(w1) ∗ · · · ∗ ξσ(m)(wm))

= θ(ξ1(x1) ∗ · · · ∗ ξm(xm)) = x1 · · ·xm.

Part 2 follows from Lemma 2.5 and the fact that the maps θ and ξi require no

arithmetic operations to compute with respect to the natural bases.

With this framework, for B a Bratteli diagram and Z a subset of the path algebra

C[Bn], to compute the sum∑

z∈Z z,

SOV Algorithm 2.1.

I. Choose m ∈ N and a subset X ⊆ (C[Bn])m such that |X| = |Z| and

Z = {x1 · · ·xm|(x1, . . . , xm) ∈ X}.

Thus, X can be thought of as a choice of factorization into at most m elements

of each element of Z.

II. For x ∈ C[Bn], let c+(x) denote the smallest integer such that x ∈ C[Bc+(x)]

and let c−(x) denote the largest integer less than or equal to c+(x) such that

x ∈ C[Bc−(x)]. Let

i+ = max{c+(xi)}, i− = min{c−(xi)},

over all ith coordinates xi of elements in X.

54


III. For each 1 ≤ i ≤ m, use quivers Qni+i− as in Section 2.3.3 to build a quiver Q

with subquivers Qi, Fi, and F .

IV. Let σ ∈ Sm and let Wi = C[Bσ(i)+ ] ∩ Centralizer(C[Bσ(i)− ]). Define

X0 = {(xσ(1), . . . , xσ(m))|(x1, . . . , xm) ∈ X} ⊆ W1 × · · · ×Wm.

For 1 ≤ i < m let

Xi = {(wi+1, . . . , wm)|(w1, . . . , wm) ∈ X0 for some w1, . . . , wi},

and let Xm = ∅.

V Define Li : Xi → A(Rσi ↑ Q;B) recursively as in Step IV preceeding Theorem

2.6.

Theorem 2.10. For B a Bratteli diagram, Z a subset of the path algebra C[Bn],

Lm := Lm(∅) =∑z∈Z

z.

Proof. This follows directly from Theorem 2.6 and Theorem 2.9.

Theorem 2.11. For B a Bratteli diagram, Z a subset of the path algebra C[Bn], we

may compute∑

z∈Z z in at most

m∑i=2

|Xi−1| dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B)

55


multiplications and

(|X0| − |X1|) +m∑i=2

(|Xi−1| − |Xi|)(dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B)− dimA(Rσ

i ↑ Q))

additions.

Proof. To compute∑z:

Stage 0: Find X0 by reordering X.

Stage 1: Compute L1 for all (w2, . . . , wm) in X1.

Stage i: Compute Li given Xi−1 and Li−1.

Stage 0 requires no operations, while stage 1 requires (|X0| − |X1|) additions. For

2 ≤ i ≤ m, we see from combining Lemma 2.5 with Theorem 2.9 that stage i requires

|Xi−1| dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B) multiplications, while the number of additions re-

quired is (|Xi−1| − |Xi|)(dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B)− dimA(Rσ

i ↑ Q)).

2.3.5 General Morphism Counts

The SOV algorithm 2.1 computes path algebra sums by first factoring each element

and then translating multiplication into maps on configuration spaces. The com-

plexity is determined by the size of the factorization sets and the dimension of the

configuration spaces.

In this section, we present methods to determine the dimension of a configuration

space. Recall from Note 2.2 that if Q is a finite subquiver of R and B a locally finite

quiver, dimA(Q ↑ R;B) = # Hom(Q ↑ R;B). In the SOV approach, B is the Bratteli

56


diagram associated to a chain of semisimple algebras and hence locally finite, so in

this section we give results to count # Hom(Q ↑ R;B).

For B a locally finite graded quiver, α, β ∈ V (B), let MB(α, β) denote the number

of paths from β to α in B. Note that for B a Bratteli diagram, α, β ∈ V (B) correspond

to irreducible representations γ, ρ and

MB(α, β) = M(γ, ρ),

as in Definition 2.7.

Theorem 2.12. Let Q,R,B be graded quivers with Q a finite subquiver of R and B

locally finite. Then

# Hom(Q ↑ R;B) =∑

φ∈Hom(V (Q)↑R;B)

∏arrowsβ→αin Q

MB(φα, φβ)

Proof. A morphism specifies the image of each vertex and each arrow. This may

be counted by first fixing the image of each vertex and counting all possible arrow

images, then varying over all possible images of V (Q).

Theorem 2.12 gives a procedure for computing # Hom(Q ↑ R;B). For a quiver

Q, let Qi denote the vertices of Q at level i. Then:

1. label each vertex αi ∈ Qi with a vertex α′i ∈ Bi such that this labeling could

extend to a map from R into B;

2. label each edge of Q from β to α by MB(α′, β′);

3. multiply the labels and sum over all possible labellings.

57


Example 2.21. Let Q1 = R1 be as in Figure 2.21. Steps 1 and 2 then give the

labelling of the figure and by Theorem 2.12,

# Hom(Q ↑ R;B) =∑α′i∈Bi

MB(α′5, α′0)MB(α′5, α

′3)MB(α′4, α

′3)MB(α′4, α

′0).

Q1 = R1 After Labelling

MB(α′4, α′0)

α′0α′3

α′4

α′5

α0α3

α4

α5

MB(α′4, α′3)

MB(α′5, α′3)

MB(α′5, α′0)

Figure 2.21

To further simplify counts, we first ‘smooth’ quivers before counting morphisms,

i.e. we remove superfluous vertices (see Corollary A.2 in Appendix A.3.1).

Example 2.22. Let Q2 = R2 be as in Figure 2.22. Then for Q1, R1 as in Figure 2.21,

Corollary A.2 gives the isomorphism

A(Q2 ↑ R2;B) ∼= A(Q1 ↑ R1;B).

To compute # Hom(Q2 ↑ R2;B), remove vertices α2 and α1, then use the labelling of

Figure 2.21.

58


Q2 = R2

α0

α1

α2

α3

α4

α5

Figure 2.22

2.3.6 Morphisms into Locally Free Bratteli Diagrams

In Section 2.3.5 we obtained general quiver morphism counting results. For locally

free Bratteli diagrams we rewrite these results in terms of the dimensions of the

corresponding subalgebras. This generalizes Stanley’s work on differential posets

[87].

Definition 2.19. A Bratteli diagram B is locally free if for each i ≥ 1, C[Bi] is

free as a module over C[Bi−1].

Example 2.23. For G a finite group, the Bratteli diagram associated to the chain

of group algebras C[G] = C[Gn] > · · · > C[G0] = C is locally free.

Let C[V (B)] denote the space of finitely supported linear combinations of vertices

of B, let Bi denote the vertices α ∈ V (B) with gr(α) = i, and let C[Bi] denote the

space of finitely supported linear combinations of vertices at level i in B. Define an

inner product 〈 , 〉 on C[V (B)] making the vertices orthonormal. As in [87] define

linear operators U and D on C[V (B)] by linearly extending the action on α ∈ Bi:

Uα =∑

γ∈Bi+1

MB(γ, α)γ,

59


Dα =∑

β∈Bi−1

MB(α, β)β,

where, by convention, if B has highest grading n,

B−1 = ∅ = Bn+1 = Bn+2 = · · · .

Note 2.4. As the vertices of B are labeled by the irreducible representations of

C[Bi], elements of C[Bi] correspond to representations of the path algebra C[Bi]. In

this context, U is induction and D restriction (see [39] Proposition 2.3.1).

We can now rewrite the sums of Theorem 2.12 as words in U and D, as in Example

2.24 below.

Example 2.24. For Q1 as in Example 2.21, we can trace each arrow on the quiver

by starting at the root, moving up four levels to vertex α4, down to vertex α3, up two

levels to vertex α5, and back down to the root. It is then easily checked that

〈D5U2DU40, 0〉 =∑

α′i∈BiMB(α′5, α

′0)MB(α′5, α

′3)MB(α′4, α

′3)MB(α′4, α

′0)

= # Hom(Q ↑ R;B).

In Corollary 2.2 we give explicit formulas for these inner products.

For α ∈ V (B) and 0 the root of B, let dα = MB(α, 0) and let di =∑

α∈Bi dαα.

Lemma 2.6.

(i) For α ∈ Bi, 〈di, α〉 = dα.

(ii) di = U i0,

Proof. Clear from definition and induction.

60


Proposition 2.1. Let B be a Bratteli diagram. Then the following properties are

equivalent:

(i) B is locally free.

(ii) For each i and all β ∈ Bi−1, there exists λi ∈ C such that

∑α∈Bi

MB(α, β)dα = λidβ.

(iii) For each i, di is an eigenvector of DU .

(iv) For each i there exists λi ∈ C with DU i0 = λiUi−10.

Proof. As this proof comes down to definitions and the fact that D is restriction (cf.

Note 2.4), we defer it to Appendix A.3.2.

Corollary 2.1. Let B be a locally free Bratteli diagram and λi the eigenvalue of DU

associated to di−1. Then λi is integral and

(i) λi =dimC C[Bi]

dimCC[Bi−1],

(ii) dimC C[Bi] =i∏

j=1

λj.

Example 2.25. For a group algebra chain C[Gn] > · · · > C[G0], the corresponding

Bratteli diagram B is locally free and

λi =dimC C[Bi]

dimC C[Bi−1]=

dimC C[Gi]

dimC C[Gi−1]= |Gi/Gi−1|.

61


Example 2.26. A Bratteli diagram is r-differential [87] if it is multiplicity-free and

DU − UD = rI. Then by induction,

DU i0 = UDU i−10 + rU i−10 = irU i−10,

hence, an r-differential poset is locally free and λi = ir.

Example 2.27. For ~r = (r0, r1, . . . ) an infinite sequence of numbers, a Bratteli

diagram is ~r-differential if it is multiplicity-free and at the ith level, DU−UD = riI.

Then by induction,

DU i0 = UDU i−10 + ri−1Ui−10 =

i−1∑j=0

rjUi−10,

hence, an ~r-differential poset is locally free and λi =i−1∑j=0

rj.

In light of Examples 2.26 and 2.27, Theorem 2.13 below generalizes Theorem 3.7

of [87] and Theorem 2.3 of [88].

Definition 2.20. Let w = wl · · ·w1 be a word in U and D and let S = {i | wi = D}.

For each i ∈ S, let ai = #{D’s in w to the right of wi}, and similarly let bi =

#{U ’s in w to the right of wi}. If bi − ai ≥ 0 for all i ∈ S, we call w an admissible

word.

Theorem 2.13. Let B be a locally free Bratteli diagram and w = DdnUun · · ·Dd1Uu1

an admissible word in U and D. Then for s =∑n

i=1 ui − di and α ∈ Bs,

〈w0, α〉 = dα∏i∈S

λbi−ai .

62


Proof. The proof comes down to inductively showing:

wk0 =∏i∈Sk

λbi−ai∑α∈Bsk

dαα.

For full details, see Appendix A.3.2

For B a locally free Bratteli diagram and Q an n-toothed quiver, Theorem 2.13

allows us to determine # Hom(Q;B).

Definition 2.21. A quiver Q is n-toothed if it consists of 2n + 1 (not necessarily

distinct) vertices γ0, . . . , γn, β1, . . . , βn and distinct arrows connecting γi−1 to βi and

γi to βi.

Example 2.28. The quiver of Figure 2.23 is an example of a 3-toothed quiver.

γ0

γ1

γ2

γ3

β1

β2

β3

Figure 2.23

Example 2.29. The quiver Q1 of Example 2.21 is 2-toothed, with γ0 = α0, γ1 =

α3, γ2 = α0, β1 = α4, β2 = α5 .

Theorem 2.14. Let B be a locally free Bratteli diagram, Q an n-toothed quiver with

vertices γi at level li, βi at level mi. Then for

w = Dmn−lnUmn−ln−1 · · ·Dm1−l1Um1−l0 ,

63


we have ∑α∈Bln−l0

〈w0, α〉 = # Hom(Q;B).

Proof. Follows from Theorem 2.12 and induction.

Corollary 2.2. Let B be a locally free Bratteli diagram, Q an n-toothed quiver with

vertices γi at level li, βi at level mi. Then for

w = Dmn−lnUmn−ln−1 · · ·Dm1−l1Um1−l0 ,

we have

# Hom(Q;B) =∑

α∈Bln−l0

〈w0, α〉 =∏i∈S

λbi−ai∑

α∈Bln−l0

dα.

Example 2.30. For Q1 as in Example 2.21, we see that

lo = l2 = 0, l1 = 3, m1 = 4, m2 = 5.

Then Bl2−l0 = B0 = 0 and by Corollary 2.2, for w = D5U2DU4,

# Hom(Q1;B) = 〈w0, 0〉 = λ4λ5λ4λ3λ2λ1.

Note that this is the inner product of Example 2.24.

We will use Corollary 2.2 for many of our complexity results in Section 2.4.

64

2.4 The Complexity of Fourier Transforms on Finite Groups

2.4 The Complexity of Fourier Transforms on Fi-

nite Groups

2.4.1 The Weyl Groups Bn and Dn

For our first application of SOV algorithm 2.1 we consider the Fourier transform of

functions on the Weyl groups of type Bn and Dn, improving upon the results of [65].

Theorem 2.15 (Theorem 2.2). For the Weyl group Bn and R a complete set of

irreducible matrix representations of Bn adapted to the subgroup chain Bn > Bn−1 >

· · · > B0 = {e},

C(Bn) ≤ TBn(R) ≤ n(2n− 1)|Bn|.

Proof. Let s1, . . . , sn denote the simple reflections for Bn, labeled according to Figure

2.24.

· · ·123n

Figure 2.24

Recall from [65] that elements in a set of minimal coset representatives forBn/Bn−1

have the following factorizations:

e, sn, sn−1sn, . . . , s1 · · · sn, s2s1 · · · sn, . . . , sn · · · s1 · · · sn,

so for Ai = {e, si} = A′i, a complete set of coset representatives is contained in

Y = {an · · · a2a1a′2 · · · a′n| ai, a′i ∈ Ai}.

65


For B the Bratteli diagram associated to the chain

C[Bn] > C[Bn−1] > · · · > C

let {C[Bi]} be the chain of path algebras associated to {C[Bi]}. Let Y = {y | y ∈ Y },

and similarly define Ai, A′i. Note that Ai, A

′i ⊆ C[Bi] ∩ Centralizer(C[Bi−2]). By

Lemma 2.3, computation of the Fourier transform of a complex function f on Bn is

equivalent to computation of

∑y∈Y

yFy =∑ai∈Aia′i∈A′i

an · · · a2a1a′2 · · · a′nFan···a2a1a′2···a′n

for Fy = Fan···a2a1a′2···a′n ∈ C[Bn−1]. We now use SOV algorithm 2.1:

I. Let X = {(an, . . . , a2, a1, a′2, . . . , a

′n, Fan···a2a1a′2···a′n)| ai ∈ Ai, a′i ∈ A′i}.

II. Note that

i+ = maxan−i+1∈An−i+1

{c(an−i+1)+} = n− i+ 1, 1 ≤ i ≤ n,

i+ = maxa′i−n+1∈A′i−n+1

{c(a′i−n+1)+} = i− n+ 1, n < i < 2n,

i− = i+ − 2, 1 ≤ i < 2n,

2n+ = max{c(Fan···a2a1a′2···a′n)+} = n− 1,

2n− = 0.

III. Glue together quivers Qni+i− in Figure 2.25 (labeled by Aj, Aj, Fy) to build the

quiver Q of Figure 2.26. The left column of Figure 2.25 shows the quivers Qni+i−

for 1 ≤ i ≤ n and the right column shows the quivers Qni+i− for n+ 1 ≤ i ≤ 2n

66


An

An−1

...

A3

A2

A1

A′2

A′3

...

A′n−1

A′n

Fy

0n n− 2

0n n− 1 n− 3

0n 3 1

0n 2

01n

0n 2

0n 13

0n n− 2

0n n− 1 n− 3

0n n− 1

Figure 2.25

Q

A′3A′4A′5 A′2

A3A4A5 A2A1

An−1 An−2

A′n−1 A′n−2

An

A′n

Fy

. . .

. . .

1234

0

1234

n− 4n− 2n− 1

n− 1

n

n− 3

n− 4n− 3n− 2

Figure 2.26

IV. Let σ ∈ S2n be the permutation reordering X so that

X0 = {(xσ(1), . . . , xσ(2n))| (x1, . . . , x2n) ∈ X}

= {(Fan···a2a1a2···an , a′2, a′3, . . . a′n, a1, a2, a3, . . . , an)}.

67


Then

X1 = {(a′2, a′3, . . . a′n, a1, a2, a3, . . . , an)| ai ∈ Ai, a′i ∈ A′i},

X2 = {(a′3, . . . a′n, a1, a2, a3, . . . , an)| ai ∈ Ai, a′i ∈ A′i},...

X2n−1 = {(an)| an ∈ An}.

Note that

|Xi−1| = |A′i| · · · |A′n||A1| · · · |An|, 2 ≤ i ≤ n,

|Xi−1| = |Ai−n| · · · |An|, n < i ≤ 2n.

By Theorem 2.11, we may compute∑yFy in at most

n∑i=2

|Xi−1| dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B) (2.4)

multiplications, with Rσi−1 ∪ Fσ(i) as in Figure 2.27.

Rσ1 ∪ Fσ(2)

A2

Rσ2 ∪ Fσ(3)

A3 · · ·

Rσn−1 ∪ Fσ(n)

An

Rσn ∪ Fσ(n+1)

A1

Rσn+1 ∪ Fσ(n+2)

A′2

· · ·

Rσ2n−1 ∪ Fσ(2n)

A′n

0n

n− 1

n− 1

n− 1 n− 2

0n 1

1

2

n− 1

0

n− 1

0

1

12 0

n− 1 12

23

0

n

n− 2n− 1

n− 1

01

n− 1

n

Figure 2.27

68


Then for Hni ,J n

i ,Kn the quivers of Figure 2.28,

dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B) = # Hom(Hn

i ↑ Q;B), 1 ≤ i ≤ n,

dimA(Rσn ∪ Fσ(n+1) ↑ Q, ;B) = # Hom(Kn ↑ Q;B),

dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B) = # Hom(J n

i−n ↑ Q;B), n+ 2 ≤ i ≤ 2n.

Hni

Ai

J ni

A′i

Kn

A′1

0n

0βn

αi−2

βi−1

αi−1

βi

0βn−1 βi−2βi−1

αi−1αi

Figure 2.28

By Lemma 2.4, A(Kn;B) ∼= C[Bn], so

dimA(Kn ↑ Q;B) = # Hom(Kn ↑ Q;B) = |Bn|

By Theorem 2.12,

# Hom(J ni ↑ Q;B) =

∑αj ,βj∈Bj

MB(βn, βi)MB(βi, βi−1)MB(βi−1, αi−2)MB(βi, αi−1)MB(αi−1, αi−2)dαi−2dβn

69


=∑

αj ,βj∈BjMB(βn, βi)MB(βi, αi−2)MB(βi, αi−1)MB(αi−1, αi−2)dαi−2

dβn

≤MB(Bi, Bi−2)∑

αj ,βj∈BjMB(βn, βi)MB(βi, αi−1)MB(αi−1, αi−2)dαi−2

dβn .

Then by Corollary 2.2,

# Hom(J ni ↑ Q;B) = MB(Bi, Bi−2)〈DnUn0, 0〉

= MB(Bi, Bi−2)|Bn|.

Simple counting in the Bratteli diagram for Bn (Lemma A.6 of Appendix A.4)

shows that MB(Bi, Bi−2) ≤ 2. Thus

# Hom(J ni ↑ Q;B) ≤ 2|Bn|.

Similarly, (using Corollary A.3 of Appendix A.4),

# Hom(Hni ↑ Q;B) ≤ 4(i− 1)

n|Bn|.

Finally, note that since a restricted product with the identity element e requires

no operations to compute (see Corollary A.1 of Appendix A.2), for all 1 ≤ i ≤ n,

|Ai| = |A′i| = 1.

Plugging in to (2.4), we may compute∑

y∈Y yFy in at most

# Hom(Kn ↑ Q;B) +n∑i=2

# Hom(Hni ↑ Q;B) +

n∑j=2

# Hom(J nj ↑ Q;B) = (4n− 3)|Bn|

70


multiplications (and fewer additions). By Lemma 2.3,

tBn(R) ≤ tBn−1(RBn−1) + 4n− 3,

and so

tBn(R) ≤ n(2n− 1).

Analogous arguments give the following result for Weyl groups of type Dn.

Theorem 2.16 (Theorem 2.3). For the Weyl group Dn and R a complete set of

irreducible matrix representations of Dn adapted to the subgroup chain Dn > Dn−1 >

· · · > D0 = {e},

C(Dn) ≤ TDn(R) ≤ n(13n− 11)

2|Dn|.

Proof. Let s1, . . . , sn denote the simple reflections for Dn, labeled according to Figure

2.29.

· · ·1

2

34n

Figure 2.29

Recall from [65] that elements in a set of minimal coset representatives forDn/Dn−1

have the following factorizations:

e, sn, sn−1sn, . . . , s3 · · · sn, s2s3 · · · sn, s1s3 · · · sn,

s1s2s3 · · · sn, s3s1s2s3 · · · sn, . . . , sn · · · s3s2s1s3 · · · sn.

71


Then for Ai = {e, si} = A′i, the proof of Theorem 2.2 shows we need only determine

# Hom(Hni ↑ Q;B), # Hom(J n

i ↑ Q;B), and # Hom(Kn ↑ Q;B), for Hni ,J n

i ,Kn the

quivers of Figure 2.28. As before,

# Hom(Kn ↑ Q;B) = |Dn|.

By Lemma A.9 and Corollary A.4 of Appendix A.4,

# Hom(J ni ↑ Q;B) ≤ 3|Dn|,

# Hom(Hni ↑ Q;B) ≤ 20(i−1)

n|Dn|,

so by Theorem 2.11 we may compute∑yFy in at most

# Hom(Kn ↑ Q;B) +∑n

i=2 # Hom(Hni ↑ Q;B) +

∑nj=2 # Hom(J n

j ↑ Q;B)

= (13n− 12)|Dn|

multiplications (and fewer additions). Then by Lemma 2.3,

tDn(R) ≤ tDn−1(RDn−1) + 13n− 12,

and so

tDn(R) ≤ n(13n− 11)

2.

72


2.4.2 The General Linear Group

Let Fq := Fpk be a finite field of characteristic p and order q = pk. Let Gln(q) denote

the matrix group Gln(Fq) and consider Gln−1(q) as a subgroup of Gln(q) under the

embedding

A→

1 0

0 A

,

for A ∈ Gln−1(q). In this section we show:

Theorem 2.17 (Theorem 2.4). For the matrix group Gln(q) and R a complete set of

irreducible matrix representations of Gln(q) adapted to the subgroup chain Gln(q) >

Gln−1(q) > · · · > {e}

C(Gln(q)) ≤ TGln(q)(R) = O(qn)|Gln(q)|.

Proof. Let P be the set of permutation matrices of Gln(q). By Proposition A.2 in

Appendix A.5, for p 6= 2,

Y = {πsi| 1 ≤ i ≤ n, (i− 1) divisible by p, π ∈ P}

contains a complete set of coset representatives for GLn(q)/Gln−1(q), where si has

form

u2 · · ·up−1u′p+1tpup+1 · · ·u2p−1u

′2p+1t2pu2p+1 · · ·uivi+1 · · · vnε,

for ε a scalar matrix, tj the permutation matrix corresponding to (j j − 1), and

uj, u′j, vj ∈ Glj(q) ∩ Centralizer(Glj−2(q)),

73


with (q − 1) possible matrices for uj and u′j, and q2 possible matrices for vj.

Let Uj (respectively U ′j, Vj, E) be the set of matrices uj (respectively u′j, vj, ε),

and let Tj = {tj}. Let Y = {y | y ∈ Y }, and similarly define Uj, U′j, Vj, Tj, E, P .

Note that

Uj, U′j, Vj, Tj ∈ C[Bj] ∩ Centralizer(C[Bj−2]),

E ∈ Centralizer(C[Bn]).

By Lemma 2.3 computation of the Fourier transform of a complex function f on

GLn(q) is equivalent to computation of:

∑y∈Y

yFy (2.5)

for Fy ∈ C[Bn−1]. The number of operations to compute (2.5) is bounded by the

number of operations to compute

∑π

∑1≤i≤np|(i−1)

∑uj ,u

′j ,

vj ,tj ,ε

πu2 · · · vi+1 · · · vnεFπu2···vi+1···vnε

=∑π

∑ε

πε∑

1≤i≤np|(i−1)

∑uj ,u

′j

vj ,tj

u2 · · · vi+1 · · · vnFπu2···vi+1···vnε. (2.6)

To compute (2.6), fix π and ε and compute:

∑1≤i≤np|(i−1)

∑uj ,u

′j

vj ,tj

u2 · · · vi+1 · · · vnFπu2···vi+1···vnε, (2.7)

then multiply by πε and sum.

To compute sums of the form (2.7):

74


I. Let

X = {(u2, . . . , up−1, u′p+1, tp, up+1, . . . , ui, vi+1, . . . , vn, Fu2···vn) | i ≤ n, i | (p−1)},

for uj ∈ Uj, u′j ∈ U ′j, vj ∈ Vj, tj ∈ Tj.

III. Glue together the quivers Qi+i− (labeled by Uj, U′j, Vj, Tj, Fu2···vn) of Figures 2.30

and 2.31 to build the quiver Q of Figure 2.32.

U2

U3

...

Up−1

U ′p+1

Tp

Up+1

Up+2

...

0n 2

0n 13

0n p− 3p− 1

0n p− 1p+ 1

0n p− 2p

0n p− 1p+ 1

0n pp+ 2

Figure 2.30

75


Ui

Vi+1

...

Vn

Fu2···vn

0n i− 2i

0n i− 1i+ 1

0n n− 2

0n n− 1

Figure 2.31

Q

U2U3Up−1TpUp+1

U′p+1

Up+2Ump−1TmpUmp+1

U′mp+1

UiVi+1Vn

Fu2···vn

. . .. . .. . .. . .01

1

2

23

p− 3

p− 2

p− 1p + 1

p + 2

mp− 3

mp− 2

mp− 1mp + 1

mp + 2

i− 2

i− 1

i

i + 1n− 1

n− 2

n

Figure 2.32

IV. Let σ ∈ Sn+m−1 be the permutation reordering X so that

X0 =: {(xσ(1), xσ(2),...,xσ(n+m−1))| (x1, . . . , xn+m−1) ∈ X}

= {(Fu2···vn , u2, u3, . . . , up−1, tp, up+1, u′p+1, up+2, ..., ui, vi+1, . . . , vn)}.

76


Then

X1 = {(u2, u3, . . . , up−1, tp, up+1, u′p+1, up+2, ..., ui, vi+1, . . . , vn)}

X2 = {(u3, . . . , up−1, tp, up+1, u′p+1, up+2, ..., ui, vi+1, . . . , vn)}

...

Xm+n−2 = {(vn)}.

By Theorem 2.11, we may compute (2.7) in at most

n∑k=2

|Xk−1| dimA(Rσk−1 ∪ Fσ(k) ↑ Q;B)

multiplications, with Rσk−1 ∪ Fσ(k) as in Figure 2.33.

Rσ1 ∪ Fσ(2)

U2

Rσ2 ∪ Fσ(3)

U3

...

Rσp ∪ Fσ(p+1)

Up+1

Rσp+1 ∪ Fσ(p+2)

U′p+1

Rσp+2 ∪ Fσ(p+3)

Up+2

...

Rσm+n−2 ∪ Fσ(m+n−1)

Vn

0n− 1

0

1

12

0n− 1 12

23

0n− 1 p− 1p

pp + 1

0n− 1

p− 1p + 1

p

0n− 1 pp + 1

p + 1p + 2

n− 1

n− 1

0

n

n− 2

Figure 2.33

77


Then for Hnj ,J n

j the quivers of Figure 2.34,

dimA(Rσk−1 ∪ Fσ(k) ↑ Q;B) = dim(Hn

j ;B) or dim(J nj ;B).

J nj

U′j

Hnj

Uj , Vj

or Tj

0n− 1

j − 2j

j − 1 0n− 1 j − 2j − 1

j − 1j

Figure 2.34

First consider the quiver Hnj of Figure 2.34, which corresponds to:

Uj, when 1 ≤ j ≤ i, p - j

Tj, when 1 ≤ j ≤ i, p | j,

Vj, when i < j ≤ n.

By Theorem 2.12,

Hom(Hnj ↑ Q;B) =

∑αi,βi∈Bi

(MB(βn−1, βj−1)MB(βj−1, βj−2)MB(αj, βj−1)

MB(αj, αj−1)MB(αj−1, βj−2)dαj−1dβn−1

)

≤MB(Glj−1(q), Glj−2(q))2|Glj−2(q)|∑

αi,βi∈Bi

(MB(βn−1, βj−1)

MB(αj, βj−1)MB(αj, αj−1)dαj−1dβn−1

).

78


By Corollary 2.2,

# Hom(Hnj ↑ Q;B) ≤MB(Glj−1(q), Glj−2(q))2|Glj−2(q)|〈Dn−1Un−jDU j 0, 0〉

= MB(Glj−1(q), Glj−2(q))2|Glj−2(q)|λjλn−1λn−2 · · ·λ1

= MB(Glj−1(q), Glj−2(q))2|Glj−2(q)| |Glj(q)||Glj−1(q)|

|Gln−1(q)|.

By [65, Lemma 5.9], M(Glj(q), Glj−1(q)) ≤ 2j−1 and |Glj(q)| ≤ qj. Thus, since

dimA(Hnj ↑ Q;B) = # Hom(Hn

j ↑ Q;B),

dimA(Hnj ↑ Q;B) ≤ 22j−4qj−2qj−1(qj − 1)|Gln−1(q)|

= 22j−4qj−2 qj−1(qj − 1)

qn−1(qn − 1)|Gln(q)|

= O(q3j−2n−2)|Gln(q)|.

Further,

|Xj| :=

#(uj, . . . , ui, vi+1, . . . , vn) 1 ≤ j ≤ i, p - j,

#(tj, . . . , ui, vi+1, . . . , vn) 1 ≤ j ≤ i, p | j,

#(vj, . . . , vn) i < j ≤ n.

In particular,

|Xj| ≤

(q − 1)i−j+1+ i−1p (q2)n−i 1 ≤ j ≤ i,

(q2)n−j+1 i < j ≤ n,

79


and so for all quivers Rσk−1 ∪ Fσ(k) of form Hn

j ,

|Xk−1| dimA(Rσk−1 ∪ Fσ(k) ↑ Q;B) = |Xj| dimA(Hn

j ,B) ≤ O(qn)|Gln(q)|.

Now consider the quiver J nj of Figure 2.34, which corresponds to Uj for all j ≤ i with

p | (j − 1):

dimA(Jj ↑ Q;B) =∑

αi,βi∈BiMB(βn−1, βj−1)MB(αj, βj−1)MB(αj, αj−2)2dαj−2

dβn−1

≤MB(Glj(q), Glj−2(q))∑

αi,βi∈BiMB(βn−1, βj−1)MB(αj, βj−1)MB(αj, αj−2)dαj−2

dβn−1

= MB(Glj(q), Glj−2(q))〈Dn−1Un−jDU j 0, 0〉

= MB(Glj(q), Glj−2(q))|Glj(q)||Glj−1(q)|

|Gln−1(q)|.

By [65, Lemma 5.9], M(Glj(q), Glj−2(q)) ≤ 22j−3qj−1. Thus,

dimA(Jj ↑ Q;B) ≤ 22j−3qj−1 qj−1(qj − 1)

qn−1(qn − 1)|Gln(q)| = O(q3j−2n−1)|Gln(q)|.

Further,

|Xj| := #(u′j, . . . , ui, vi+1, . . . , vn) ≤

(q − 1)i−j+1+ i−1p (q2)n−i j 6= n

(q − 1) j = i = n

and so for all quivers Rσk−1 ∪ Fσ(k) of form J n

j ,

|Xk−1| dimA(Rσk−1 ∪ Fσ(k) ↑ Q;B) = |Xj| dimA(Jj ↑ Q;B) ≤ O(qn)|Gln(q)|.

80


Thus, by Theorem 2.11, we may compute (2.7) in at most O(qn)|Gln(q)| operations.

In order to compute (2.5) we must consider multiplication by ε and by π. Let

Fε ∈ C[Bn]. To compute∑

ε εFε, note that ε is a scalar matrix, and so a single

multiplication εFε requires at most n2 multiplications. Then∑

ε εFε requires at most

(q − 1)n2 < O(qn)|Gln(q)| operations.

Finally, consider multiplication by π ∈ P . Let Fπ ∈ C[Bn]. To compute∑

π πFπ,

note that π is a permutation matrix, and so every row and column contains exactly

one nonzero entry, and that entry is 1. Then a single multiplication πFπ requires no

multiplications, and so∑

π πFπ does not add to the complexity.

Now suppose p = 2. By Theorem A.5 in Appendix A.5,

Y = {πsi| 1 ≤ i ≤ n, (i− 1) divisible by p, π ∈ P}

contains a complete set of coset representatives for Gln(q)/Gln−1(q), where si is of

form

a3b2c3a5b4c5 · · · aibi−1civi+1 · · · vn,

for aj, bj, cj ∈ Glj(q)∩Centralizer(Glj−2(q)) with (q− 1) possible matrices for aj and

bj, q2 possible matrices for vj, and cj completely determined by aj and bj−1. The

same arguments as in the p 6= 2 case then yield the quiver Q of Figure 2.35, from

which it is clear that analogous arguments show that the Fourier transform may be

computed in O(qn)|Gln(q)| operations.

81


Q

b2c3

a3

b4c5

a5

bi−1ci

ai

vi+1vn

Fs′

. . .. . .01

1

2

3

34

5

i− 3i− 2i− 1i

i− 2ii + 1n− 1

n− 2n− 1

n

Figure 2.35

2.4.3 Generalized Symmetric Group Case

We next give a general result (Theorem 2.18) to find efficient Fourier transforms on

groups with special subgroup structure. As the proof follows the same structure of

the proofs of Theorems 2.24, 2.29, and 2.4, we leave it as an exercise.

Suppose

Gn > Gn−1 > · · · > G0 = e,

is a chain of subgroups with subsets Ai ⊆ Gi such that

(1) A1 = G1

(2) Gi = A2 · · ·AiGi−1 for 2 ≤ i ≤ n.

(3) Ai commutes with Gi−2.

Let B be the Bratteli diagram associated to the chain

C[Gn] > C[Gn−1] > · · · > C,

82


and let {C[Bi]} be the chain of path algebras associated to {C[Gi]}. Let

MB(Gi, Gj) := maxMB(αi, αj)

over all αi ∈ Bi, αj ∈ Bj and let |Gi| denote the number of conjugacy classes of Gi;

equivalently, the number of irreducible representations in a complete set of inequiva-

lent irreducible representations of Gi.

Theorem 2.18. Let Gi, Ai be as described above. Then the Fourier transform of a

complex function on Gn may be computed at a complete set R of irreducible represen-

tations of Gn adapted to the chain

Gn > Gn−1 > · · · > G0 = e

in at most

|Gn|n∑k=1

k∑i=2

MB(Gi−1, Gi−2)2|Gi−2||Gi||Gi−1|

|Gk−1||Gk|

k∏j=i

|Aj|

operations.

Note 2.5. Theorem 2.18 is a refinement of Theorem 3.1 of [64]: rather than consid-

ering the maximum length of an element in a set of coset representatives, we build

the coset representatives up using sets Ai of smaller size.

Note 2.6. For Gi = Si, this theorem gives an efficient algorithm for the computation

of the Fourier transform of a function on the symmetric group by letting A1 = {e}

and Ai = {e, ti−1} for 2 ≤ i ≤ n.

83


2.4.4 The Complexity of Fourier Transforms on Homoge-

neous Spaces

We next consider the Fourier transform of a function on a homogeneous space, a

special case of harmonic analysis on groups. This can be viewed as a coset space

G/K, so a Fourier transform on a homogeneous space is a Fourier transform of the

space of functions on G/K or, equivalently, of the space of associated right-K in-

variant functions on G. See [63, 65] for further background on Fourier transforms on

homogeneous spaces and some of their applications.

Definition 2.22. Let G be a finite group with subgroup K and let f be a complex-

valued function on G/K. The Fourier transform of f at a K-adapted represen-

tation ρ of G, denoted f(ρ)K or a K-adapted set R of matrix representations of G,

denoted FKR f is the Fourier transform of the right K-invariant function f : G → C

defined by

f(g) =1

|K|f(gK).

Note that f(ρ) is zero unless the representation space, Vρ, contains a nontrivial

K-invariant vector. Such a representation is said to be class 1 relative to K, and

we could restrict to class 1 representations if desired.

Definition 2.23. Let G be a finite group with subgroup K and let R be a set of

representations of G.

(i) The arithmetic complexity of a Fourier transform on R, denoted TG/K(R),

is the minimum number of arithmetic operations needed to compute the Fourier

transform of f on R via a straight-line program for an arbitrary complex-valued

function f defined on G/K.

84


(ii) The reduced complexity, denoted tG/K(R), is defined by

tG(R) =1

|G/K|TG/K(R).

Note that the complexity always satisfies the inequalities

|G/K| − 1 ≤ TG/K(R) ≤ |G/K|2.

Further, the proof of Lemma 2.3 gives an analogous result for the case of homoge-

nous spaces: for H a subgroup of G, R a complete H-adapted set of inequivalent

irreducible representations of G, Y ⊆ G a set of coset representatives, and Y the

corresponding subset of the path algebra under the isomorphism of Lemma 2.1,

tG/K(R) ≤ tH/K(RH) +mG/K(R, Y ,H).

With this result, we obtain complexity results for homogeneous spaces as in the

group case.

Let G be a group with chain of subgroups G = Gn > Gn−1 > · · · > G0. For f a

function on Gn that is right Gn−k-invariant, the corresponding element∑

s∈Gn f(s)s

in C[Gn] is invariant under right multiplication by elements of C[Gn−k]. In partic-

ular, the elements Fy in the proofs of Section 2.4 are C[Bn−k]-invariant, so as in

[65, Theorem 6.2] the nonzero coefficients of Fy correspond to paths passing through

1n−k. Using the SOV approach as in the proofs of Section 2.4, the final quiver in the

construction of Q, Qi+i− , corresponding to Fy now has form as in Figure 2.36, with

Hom(Qi+i− ;B) consisting only of morphisms sending αn−k to 1n−k.

85


0αn αn−1

αn−k

Figure 2.36

Then the proofs of Section 2.4 extend to the following results for homogenous

spaces:

Theorem 2.19 (Theorem 2.5). For the homogenous space Bn/Bn−k of the Weyl

group Bn and R a complete set of irreducible matrix representations of Bn adapted to

the subgroup chain Bn > Bn−1 > · · · > {e},

C(Bn/Bn−k) ≤ TBn/Bn−k(R) ≤ k(4n− 2k − 1)|Bn||Bn−k|

.

Theorem 2.20. For the homogenous space Dn/Dn−k of the Weyl group Dn and R a

complete set of irreducible matrix representations of Dn adapted to the subgroup chain

Dn > Dn−1 > · · · > {e},

C(Dn/Dn−k) ≤ TDn/Dn−k(R) ≤ k(26n− 13k − 11)

2

|Dn||Dn−k|

.

Theorem 2.21. For the homogenous space Gln(q)/Gln−k(q) of the general linear

group Gln(q) and R a complete set of irreducible matrix representations of Gln(q)

adapted to the subgroup chain Gln(q) > Gln−1(q) > · · · > {e},

C(Gln(q)/Gln−k(q)) ≤ TGln(q)/Gln−k(q)(R) ≤ O(qn)|Gn||Gn−k|

.

86

2.5 Extension to Semisimple Algebras

As in Section 2.4.3, suppose

Gn > Gn−1 > · · · > G1 = e,

is a chain of groups with subsets Ai ⊆ Gi such that

(1) A1 = G1

(2) Gi = A2 · · ·AiGi−1 for 2 ≤ i ≤ n.

(3) Ai commutes with Gi−2.

Theorem 2.22. Let Gi, Ai be as above. For the homogeneous space Gn/Gn−k and R

a complete set of irreducible matrix representations of Gn adapted to the chain

Gn > Gn−1 > · · · > G1 = e

C(Gn/Gn−k) ≤ TGn/Gn−k(R) ≤n∑

j=n−k+1

j∑i=2

MB(Gi−1, Gi−2)2|Gi−2||Gi||Gi−1|

|Gj−1||Gj|

j∏l=i

|Al|

operations.


Recall that an algebra A is semisimple if A decomposes as a direct sum of simple

algebras:

A ∼=⊕λ∈Λ

Mdλ(C),

for Λ a finite index set.

87


In this section we extend the SOV approach to general semisimple algebras and

apply it to the Brauer and BMW algebras. Note that although we considered complex

group algebras in Sections 2.1-2.4, for any field F , we may define the group algebra of

a group G over F to be the F -linear span of elements in G. Then for a parameter q, the

group algebra of the symmetric group over C(q) is a subalgebra of the Brauer algebra:

C(q)[Sn] < Brn, while the BMW algebra is a “deformation” of the Brauer algebra.

As such, these results are natural extensions of Fourier transforms of functions on the

symmetric group.

2.5.1 Background: The Brauer Algebra

First note that an element in the symmetric group Sn can be realized as a diagram

on 2n points, consisting of two rows of n points each, with each point in the top row

connected by an edge to exactly one point in the bottom row (see Figure 2.37). For

two elements x, y in Sn, the product xy is the concatenation of the two diagrams: to

compute the product xy, place the diagram for x on top of the one for y and trace the

edges from top to bottom (note that we consider multiplication from left to right).

1 2 3 4

1 2 3 4

Figure 2.37: (1324)

88


=

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Figure 2.38: (1324)*(143)=(23)

The simple transpositions {ri = (i i + 1) | 1 ≤ i ≤ n − 1} form a generating set

for the symmetric group.

Elements of the Brauer monoid, Brn, are realized by generalizing symmetric group

diagrams: consider diagrams on 2 rows of n points each, with edges connecting pairs

of points regardless of row and each point part of exactly one edge. Multiplication

may again be realized as concatenation of diagrams. Note that in some cases, this

introduces a closed loop. A parameter q is used to keep track of the number of closed

loops: for two diagrams x, y ∈ Brn, let c denote the number of closed loops in the

multiplication xy and let z be the diagram of this product with the closed loops

removed. Then xy = qcz.

=x

yq z

Figure 2.39: xy = q1z

Two Brauer diagrams d1 and d2 are equivalent if they differ only in the number

of closed loops, i.e., if when q = 1, d1 = d2. For example, for x, y, z as in Figure

2.39, the product xy is equivalent to z. The Brauer monoid, Brn consists of the

89


set of equivalence classes of such diagrams and is generated by the set of elements

{ri, ei | 1 ≤ i ≤ n − 1} (see Figure 2.40). Note that the symmetric group Sn is

generated by the transpositions {ri | 1 ≤ i ≤ n− 1} and so C(q)[Sn] < Brn.

. . . . . .

i i+ 1

ri

. . . . . .

i i+ 1

ei

Figure 2.40: ri, ei ∈ Brn

The Brauer algebra, Brn, is the C(q)-algebra with basis Brn. Equivalently (see,

e.g., [5]), Brn has algebraic presentation given by generating set

{ri, ei | 1 ≤ i ≤ n− 1},

and relations:

(1) r2i = 1, (2) rirj = rjri, riej = ejri, eiej = ejei, |i− j| > 1

(3) e2i = qei, (4) eiri = riei = ei,

(5) riri+1ri = ri+1riri+1, (6) eiei+1ei = ei, ei+1eiei+1 = ei+1,

(7) riei+1ei = ri+1ei, (8) ei+1eiri+1 = ei+1ri.

In [93], Wenzl showed that the Brauer algebra, Brn(q), is a semisimple algebra

over C(q). In fact, replacing q by α ∈ C, Brn(α) is semisimple for all but finitely

many integers α [83].

90


2.5.2 Generalized SOV Approach

For A a finite-dimensional semisimple algebra with basis D, recall that to compute the

Fourier transform of f =∑d∈D

f(d)d we consider computation of the sums∑d∈D

f(d)ρ(d)

for a complete set of inequivalent irreducible representations ρ of A. By Lemma 2.2,

for a chain of subalgebras A = An > · · · > A0 this is equivalent to the computation

of ∑f(d)d

in the path algebra corresponding to the chain, by expressing d in coordinates with

respect to a system of Gel’fand-Tsetlin bases.

Recall from Lemma 2.3 that for a finite group G with subgroup H, computation

of a Fourier transform on G is equivalent to computation of

∑y∈Y

yFy,

for Y a set of coset representatives for G/H.

While there is no notion of coset representatives for a general chain of semisimple

algebras, factorization through the chain still gives the result of Lemma 2.3 and so

the SOV approach extends to this general case.

Let A be a finite-dimensional semisimple algebra. Recall that TA(R) is defined to

be the minimum number of operations to compute the Fourier transform of f with

respect to a set of representations R of A. Recall also that

tA(R) :=1

dim(A)TA(R).

91


Let C(A) = minR{TA(R)}, where R varies over all complete sets of inequivalent

irreducible matrix representations of A. As in the group case, direct computation of

a Fourier transform on A gives C(A) ≤ dim(A)2.

For A1 a subalgebra of A let B be the Bratteli diagram of the chain A > A1 > A0,

with path algebra chain C[BA] > C[BA1 ] > C[BA0 ]. For Y a set of elements of A and

Fy arbitrary elements of C[BA1 ], let

mA(R, Y,An−1) =1

dim(A)×

minimum number of operations required to compute∑y∈Y

yFy in a system of Gel’fand-Tsetlin bases for B.

Lemma 2.7. Let A1 be a subalgebra of A, R a complete A1-adapted set of inequivalent

irreducible matrix representations of A, D a basis for A, and D′ a basis for A1. Let

Y ⊆ D be a set of elements of A such that for all d ∈ D, d = yd′ for y ∈ Y , d′ ∈ D′.

Let B be the Bratteli diagram of the algebra chain A > A1 > A0 with corresponding

path algebra chain C[BA] > C[BA1 ] > C[BA0 ]. Then

tA(R) ≤ tA(RAn−1) +mA(R, Y,An−1).

With Lemma 2.7, we use SOV algorithm 2.1 to compute the Fourier transform of

functions on semisimple algebras.

2.5.3 The Brauer Algebra

Theorem 2.23. The Fourier transform of an element f in the Brauer algebra Brn

may be computed at a complete set R of irreducible matrix representations of Brn

92


adapted to the chain of algebras

Brn > Brn−1 > · · · > Br0 = C(q)

in at most (4n2 − n+ 4) dim(Brn) = (4n2 − n+ 4)(2n− 1)!! operations.

Proof. A basis for the Brauer algebra consists of Brauer monoid elements: diagrams

d ∈ Brn on 2n points as described in Section 2.5.1, generated by the elements ei and

ri, 1 ≤ i ≤ n− 1. Consider the sets

R = {1, r1 · · · rn−1, r2 · · · rn−1, . . . , rn−1},

ER = {rj · · · ri−1ei · · · en−1 | 1 ≤ i ≤ n− 1, 1 ≤ j ≤ i− 1}

Then for Y = R ∪ER, Y satisfies the conditions of Lemma 2.7. To see this, first

note that R ⊆ Sn forms a complete set of coset representatives for Sn/Sn−1 [63] and

so we need only show that for any d ∈ Brn − Sn, d = yd′, for y ∈ Y , d′ ∈ Brn−1. We

view diagrams in Brn−1 as elements of Brn by adding a point to the end of the top

and bottom rows and connecting these two points with an edge.

Each element of ER has exactly one horizontal edge in the bottom row, and this

edge connects the last two points in this row. Further, each element of ER has exactly

one horizontal edge in the top row, and each possible such edge corresponds to an

element of ER. As an example, see Figure 2.41.

Let d ∈ Brn − Sn. Then d has at least one horizontal edge, e, in the top row.

Choose the element, y, of ER with edge e. Then this determines an element d′, in

Brn−1 with d = yd′. For an example, see Figure 2.42.

93


e1e2e3 r1e2e3 r1r2e3 e2e3 r2e3 e3

Figure 2.41: ER in Br4

=

Figure 2.42: d = r1e2e3d′

Thus, Y satisfies the conditions of Lemma 2.7 and for Ai = {id, ri, ei},

Y ⊆ {a1a2 · · · an−1 | ai ∈ Ai}.

Note that ri, ei ∈ Bri ∩ CentralizerBri−2.

For B the Bratteli diagram associated to the chain

Brn > Brn−1 > · · · > Br0 = C(q),

let {C[Bi]} be the chain of path algebras associated to {Bri}. Let Y = {y | y ∈ Y }

and similarly define Ai. Note that Ai ⊆ C[Bi]∩Centralizer(C[Bi−2]). By Lemma 2.7,

the complexity of the computation of a Fourier transform of a complex function f on

Brn is bounded by the complexity of computation of:

∑ai∈Ai

a1 · · · an−1Fa1···an−1 .

Using the SOV approach, we may compute this sum in at most

94


n∑i=2

|Xi−1| dimA(Rσi−1 ∪ Fσ(i) ↑ Q;B) (2.8)

multiplications, where for Hni the quiver of Figure 2.43, dimA(Rσ

i−1 ∪ Fσ(i) ↑ Q;B)=

# Hom(Hni ↑ Q;B).

Hni

Ai

0βn−1 βi−2βi− 1

αi−1αi

Figure 2.43

By Corollary A.5 in Appendix A.4.3,

# Hom(Hni ↑ Q;B) ≤ 16i− 17

2n− 1dim(Brn).

Then by Lemma 2.7 and Equation 2.8,

tBrn(R) ≤ tBrn−1(RBrn−1) + 2n∑i=2

16i− 17

2n− 1

= tBrn−1(RBrn−1) + 2(n− 1)(8n− 1)

(2n− 1)

≤ tBr1(RBr1) + 2n∑i=2

(n− 1)(8n− 1)

(2n− 1)

≤ tBr1(RBr1) + (4n2 − n+ 3)

= 4n2 − n+ 4.

(2.9)

95


2.5.4 The BMW Algebra

The BMW algebra is defined in a similar manner to the Brauer algebra (see Section

3.3.3 of Chapter 3). The Bratteli diagram for the BMW algebra is identical to that

of the Brauer algebra. Further, a natural basis, Bn = {Td | d ∈ Brn} for the BMW

algebra is indexed by Brauer monoid elements. As such, Theorem 2.23 extends to

the BMW algebra:

Theorem 2.24. The Fourier transform of an element f in the BMW algebra BMWn

may be computed at a complete set R of irreducible matrix representations of BMWn

adapted to the chain of algebras

BMWn > BMWn−1 > · · · > BMW0

in at most (4n2 − n+ 4) dim(BMWn) = (4n2 − n+ 4)(2n− 1)!! operations.

2.5.5 General Result

We next give a general result (Theorem 2.25) to find efficient Fourier transforms on

a finite dimensional semisimple algebra A with special subalgebra structure. This is

a generalization of Theorem 2.18.

Suppose

A = An > An−1 > · · · > A0,

is a chain of subalgebras of A with subsets Bi ⊆ Ai such that

(1) B1 = A1

(2) Ai = B2 · · ·BiAi−1 for 2 ≤ i ≤ n.

96


(3) Bi commutes with Ai−2.

Let B be the Bratteli diagram associated to the chain

An > An−1 > · · · > A0,

and let {C[Bi]} be the associated chain of path algebras. Let

MB(Ai, Aj) := maxMB(αi, αj)

over all αi ∈ Bi, αj ∈ Bj and let |Ai| denote the number of irreducible representations

in a complete set of inequivalent irreducible representations of Ai.

Theorem 2.25. Let Ai, Bi be as described above. Then the Fourier transform of an

element f ∈ A may be computed at a complete set R of irreducible representations of

An adapted to the chain

An > An−1 > · · · > A0

in at most

dim(An)n∑k=1

k∑i=2

MB(Ai−1, Ai−2)2|Ai−2|dim(Ai)

dim(Ai−1)

dim(Ak−1)

dim(Ak)

k∏j=i

|Bj|

operations.

97

Chapter 3

Random Walks on the

Birman-Murakami-Wenzl Algebra

3.1 Introduction

Markov Chain Monte Carlo (MCMC) methods are used widely to simulate and study

stochastic systems. While the idea of using probabilistic simulation to understand

the behavior of a system predates the existence of computers, it was not until after

World War II that these methods began to take hold, as a focus on the development

of nuclear weapons combined with access to a device that could carry out large-scale

simulations enabled researchers to estimate the behavior of large collections of atomic

particles through sampling (see [79] for a history of MCMC methods).

The main idea behind MCMC methods is that they provide algorithms for sam-

pling from a probability distribution π by simulating some suitable random dynamics

that converge to π. These dynamics are phrased in terms of Markov chains, stochas-

tic processes that describe transitions between states. We describe Markov chains in

98

3.1 Introduction

more detail in Section 3.2.1, and further background can be found in many standard

probability texts (see e.g. [33, 57]).

One of the earliest and most widely used MCMC method is the Metropolis algo-

rithm. First introduced by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller

to analyze the hard spheres model [68], a model of nonoverlapping molecules, the

Metropolis algorithm gives a method for sampling from a probability distribution π

by modifying an existing Markov chain (K(x, y))x,y∈X to produce a new Markov chain

(M(x, y))x,y∈X with stationary distribution π. This proves particularly useful for sim-

ulating configurations of particles with an associated energy (e.g., the influence that

neighboring particles exert on each other). Metropolis et al. noted that sampling from

the uniform distribution for such systems is highly unrealistic, as physical laws dictate

that low-energy configurations are far likelier to occur in practice. The Metropolis

algorithm instead provided a way to sample from the Boltzmann distribution, which

weights configurations by their likelihood of being observed. Later applications of

the Metropolis algorithm include the simulation of Ising models, initially developed

to model a ferromagnet but (surprisingly) also of use in image analysis, and Gibbs

sampling [10, 34]. See [58] for additional applications.

The Metropolis algorithm has the advantage of being straightforward to construct

and implement, despite the often complicated nature of the system under study.

However, in analyzing the rate of convergence to π (the mixing time) rigorous bounds

are often dependent on the specific situation (see [74] for a review of the existing

literature for spin systems alone). Further, these methods are most often examples

of random update Markov chains in that the process involved is that of selecting a

site or set of sites to update at random. In other words, if the process of updating a

99

3.1 Introduction

particular site i can be characterized by the application of a Markov chain Mi, then

a random scan has form Mil . . .Mi2Mi1 for a sequence {il} chosen at random.

A more intuitively appealing and often more frequently used method in experi-

mental work is that of a systematic scan Markov chain, a method to cycle through and

update the sites in a deterministic order, i.e., a deterministic choice of the sequence

{il}. For example, consider the toy problem presented in [25]: for 0 < θ ≤ 1 try to fill

n empty spaces x1, . . . , xn with 0’s and 1’s so that the probability of a configuration

with m 1’s is θn−m(1− θ)m. A random scan would pick spaces to fill at random, and

take on the order of 14n log n steps to converge, while a systematic approach would

work from left to right, filling each space xi with a θ-coin toss, and require order n

steps. For examples of ‘real’ applications of systematic scans, see [10, 30, 52, 74].

Systematic scans often prove harder to analyze than random update chains due

to the need to study the entire scan’s effect and so understand all intermediary steps.

Recently, results have begun to bring the upper bounds on mixing time for systematic

scans closer to those on random scans [48, 73, 75], or in some cases have shown these

bounds to be the same [25].

In [25], Diaconis and Ram consider the problem of comparing systematic scanning

techniques with random scanning techniques in the context of generating elements of

a finite Coxeter group W . They use the Metropolis algorithm to produce systematic

scans for generating elements of W . To analyze these scans, they translate the chains

into left multiplication operators in the Iwahori-Hecke Algebra corresponding to W ,

an algebra which can be thought of as a deformation of W . For a finite Coxeter

group W generated by simple reflections s1, . . . , sn, the corresponding Iwahori-Hecke

100

3.1 Introduction

algebra is the algebra with basis {Tw | w ∈ W} and multiplication

TsiTw =

Tsiw l(siw) = l(w) + 1,

(q − 1)Tw + qTsiw, l(siw) = l(w)− 1,

for q an indeterminate and l(w) the minimum number of reflections needed to express

w [18].

Hecke algebras arise naturally in the extension of Schur-Weyl duality to general

centralizer algebras. For V ∼= Cn, there exist natural actions of the groups G =

Gln(C) and S = Sk on the k-fold tensor space V ⊗k. In [85] Schur found that these

actions are full centralizers of each other and so V ⊗k decomposes as a direct sum of

simple C[G]-modules tensored with simple C[S]-modules.

This extends to more general algebras through the Double Centralizer Theorem

(see e.g. [42] for a statement). In particular, for G = Gln(Fq) and B the subgroup of

upper triangular matrices, let L(G/B) be the space of right B-invariant functions on

G. Then for H the Hecke algebra of the symmetric group, L(G/B) decomposes as

a direct sum of simple G-modules tensored with simple H-modules [18]. Even more

generally, this statement holds for G a finite Chevalley group, B its Borel subgroup,

and H the Hecke algebra corresponding to a finite Coxeter group W . A restatement

shows that the Hecke algebra is in Schur-Weyl duality with the quantum universal

enveloping algebra of the general linear group [50]. This duality has motivated study

of the representation theory of the Hecke algebra, and more generalized versions of

these algebras (see e.g. [1, 29, 41, 92, 78]).

More relevant for this thesis is an alternative definition of the Hecke algebra in

terms of braids. The thesis [38] gives a thorough introduction to braids and their

101

3.1 Introduction

relationship with the Hecke algebra. Let b1, . . . , bn ∈ R with b1 < · · · < bn. An

n-strand braid is a disjoint union of n smooth curves in R3 connecting the points

{(b1, 1, 0), (b2, 1, 0), . . . , (bn, 1, 0)} with {(b1, 0, 0), (b2, 0, 0), . . . , (bn, 0, 0)} so that they

intersect each parallel plane y = t as t ranges between 0 and 1 only once. A braid

can be represented by its 2-dimensional projection, its braid diagram, and connecting

the top strands to the bottom strands of a braid diagram gives rise to a link. Two

links are isotopic if they are related by a sequence of Reidemeister moves (defined

in Section 3.3.3), and, in fact, every isotopic oriented link can be represented by

the closure of a braid [38]. The braid group has a natural presentation in terms

of generators Tr1 , . . . , Trn−1 corresponding to certain braid diagrams. Remarkably,

adding a quadratic relation to this presentation yields the Hecke algebra, and in this

case the elements Tri are exactly those of Figure 3.4 in Setion 3.3.3.

Under this definition of the Hecke algebra there is a natural generalization to

the BMW algebra. By generalizing the notion of an n-strand braid to allow any

two points in {(b1, 1, 0), (b2, 1, 0), . . . , (bn, 1, 0)} ∪ {(b1, 0, 0), (b2, 0, 0), . . . , (bn, 0, 0)} to

be connected, we have the definition of an n-tangle, which gives rise to the idea of

a tangle diagram by considering its two-dimensional projection. We define tangle

diagrams in detail in Section 3.3.3. As with the algebra associated to braid diagrams,

an algebra is naturally associated to these tangle diagrams. Defined independently

as the Kauffman tangle algebra by Murakami [70] and algebraically by Birman and

Wenzl [7], it was shown in an unpublished paper by Wasserman [69] that these two

notions are equivalent, giving rise to the single BMW algebra.

Further, just as the Hecke algebra is in Schur-Weyl duality with the quantum

universal enveloping algebra of the general linear group, the BMW algebra is in

102

3.2 Preliminaries: Probability Theory

Schur-Weyl duality with the quantum universal enveloping algebra of the orthogonal

group [45]. This duality has motivated the study of the representation theory of the

BMW algebra and its generalizations [16, 21, 22, 45, 56].

A third perspective would note that just as the Hecke algebra is a q-analogue of

the symmetric group, the BMW algebra is a q-analogue of the Brauer algebra, as

defined in 2.5.1.

As discussed above, in [25], Diaconis and Ram consider the problem of systemati-

cally generating elements of a finite Coxeter group W . In terms of the group algebra

C[W ], this problem is equivalent to generating elements of the basis W of C[W ]. We

extend these ideas to the BMW algebra. The Metropolis algorithm in this context

gives rise to systematic scanning strategies for generating basis elements via multipli-

cation of generators. As the diagrams forming the BMW monoid basis of the BMW

algebra are tangles, scanning strategies for generating BMW monoid elements have

applications arising in physics: random generation of links and tangles has been of

use in [28, 60, 96]. As in [25], our algorithm gives rise to a natural random walk, in

this case on the BMW and Brauer monoids, defined in Sections 3.3.2 and 3.3.3. We

translate the random walk into a multiplication operator in the BMW algebra and

develop a trace form to study it. This enables the use of tools from representation

theory and Fourier analysis to analyze the time to stationarity of such walks.


Background on Markov chains can be found in many standard probability texts (see

eg [33]). The book of Levin, Peres, and Wilmer [57] gives a particularly thorough

introduction to Markov chains, including classification of states and the Metropolis

103


algorithm, while [25] gives a concise introduction to the probabilistic background

needed for this chapter. We will follow the notation and outline of [25].

3.2.1 Markov Chains

A finite Markov chain with state space X is a process that moves among states in X

such that the conditional probability of moving from state x to state y is independent

of the preceding sequence of states. More formally:

Definition 3.1. A Markov chain on a finite set X is a matrix K = (K(x, y))x,y∈X

such that K(x, y) ∈ [0, 1] and for all x ∈ X,

∑y∈X

K(x, y) = 1.

We call X the state space.

Note that K(x, y) gives the probability of moving from x to y in one step, while

Km(x, y) gives the probability of moving from x to y in m steps.

Definition 3.2. A Markov chain K is irreducible if for each x, y ∈ X, there exists

an integer m such that Km(x, y) > 0. Let T (x) denote the minimum number of steps

for the chain to return to x, i.e. the minimum t such that Kt(x, x) > 0. Then K is

aperiodic if

gcdx

(T (x)) = 1.

Note that if K is irreducible and aperiodic, there exists an integer r such that

Kr(x, y) > 0 for all x, y ∈ X [57, Proposition 1.7].

104


Definition 3.3. A Markov chain is reversible if there exists a probability distribution

π : X → [0, 1] such that for all x, y ∈ X,

π(x)K(x, y) = π(y)K(y, x).

We call π the stationary distribution of K.

An irreducible, aperiodic, reversible Markov chain K converges to its stationary

distribution:

liml→∞

Km(x, y) = π(y).

The Metropolis construction introduced in Section 3.2.2 produces a reversible

Markov chain with a chosen stationary distribution. Our interest is in the time to

stationarity of such chains.

Definition 3.4. Let Kmx denote the probability distribution Km(x, ·). The total vari-

ation distance from Kmx to π is

|Kmx − π|TV := max

A⊆X|∑y∈A

Km(x, y)− π(y)|.

For L2(π) the space of functions f : X → R, equipped with the inner product

〈f, g〉2 =∑

f(x)g(x)π(x),

the total variation distance is bounded by the L2(π) norm:

105


Lemma 3.1. [25, Lemma 2.3] For f ∈ L2(π),

|f |2TV ≤1

4‖f/π‖2

2,

where f/π(x) = 0 if π(x) = 0.

3.2.2 The Metropolis Algorithm

Given a symmetric Markov chain P and a probability distribution π, the Metropo-

lis algorithm modifies P to produce a reversible Markov chain M with stationary

distribution π:

M(x, y) =

P (x, y) if x 6= y and π(y) ≥ π(x),

P (x, y)π(y)

π(x)if x 6= y and π(y) < π(x),

P (x, x) +∑

π(z)<π(x)

P (x, z)

(1− π(z)

π(x)

)if x = y.

As an example, in [25], Diaconis and Ram consider the Markov chain that arises

from applying the Metropolis algorithm to the usual random walk on the symmetric

group based on a generating set. We consider this chain in further detail in Section

3.4.

The Metropolis algorithm yields a reversible Markov chain with stationary distri-

bution π; however, irreducibility and aperiodicity is not guaranteed. In particular,

the Markov chains we consider in Section 3.4 are aperiodic but not irreducible. To

analyze these chains we consider their closed communication classes.

Definition 3.5. Let K be a Markov chain with state space X. For x, y ∈ X, y is

106


accessible from x, denoted x→ y, if x can reach y in finitely many steps. We say x

communicates with y, denoted x↔ y, if x→ y and y → x. The equivalence classes

under the relation↔ are the communication classes of K. A communication class

C is closed if for x ∈ C and for all y /∈ C, y is not accessible from x.

Note that studying the time to stationarity of a reversible, aperiodic Markov chain

K reduces to studying the time to stationarity of the closed communication classes

of K.

3.2.3 Systematic Scans

The Metropolis algorithm, in the context of generating elements of a group, provides

systematic and random scanning strategies. For example, for each generator ri =

(i i+ 1) of Sn, let

Pi(x, y) =

1 if y = rix,

0 else.

Then for lS the length function on words in Sn, let π be the probability distribution

π(x) =θ−lS(x)∑

w∈Sn

θ−lS(w).

The Metropolis algorithm construction then produces Markov chains M1,M2, . . . ,

Mn−1 corresponding to multiplication by the generators r1, · · · , rn−1. For an explicit

description see Section 3.4.

A choice of infinite sequence {il}∞l=1 gives a scanning strategy:

· · ·MilMil−1· · ·Mi1 .

107


For Mi reversible, each with stationary distribution π, the following systematic scans

produce reversible Markov chains with stationary distribution π (see, eg [25]):

1

n− 1

n−1∑i=1

Mi (random scan),

M1M2 · · ·Mn−1Mn−1 · · ·M2M1 (short systematic scan),

(M1 · · ·Mn−1Mn−1 · · ·M1) · · · (M1M2M2M1)(M1M1) (long systematic scan).

While such scanning strategies may seem intuitive for sampling from π, they

have proven difficult to analyze in many situations. In the context of generation of

Coxeter group elements, Diaconis and Ram [25] show that convergence of the short

systematic scan for the distribution π above, with lS replaced by the length function

on the Coxeter group coming from writing words as a product of simple reflections,

occurs in the same number of steps as that of a random scan, i.e., choosing a random

sequence of indices {i`}∞`=1. However, results for different scanning techniques or

probability distributions remain open. In the context of graph colorings, Dyer et al.

compare systematic scans with random scans for sampling proper q-colorings of paths

for q ≥ 4, in which a vertex is assigned a new color c only if none of its neighbors are

colored by c [30]. However, results for more general graphs have resisted analysis.

Fishman [34] gives an overview of scanning strategies, while Diaconis and Saloff-

Coste’s survey [27] provides further applications of the Metropolis algorithm.

108

3.3 Preliminaries: Semisimple Algebras


3.3.1 Fourier Inversion and Plancherel

Recall from Chapter 2 that for A a semisimple algebra with basis {ai}i∈I , f =∑i∈I

f(ai)ai ∈ A, and ρ : A → Md(C) a matrix representation of A, the Fourier

transform of f at ρ is the matrix sum:

f(ρ) =∑i∈I

f(ai)ρ(ai).

Random walks on groups are frequently studied using Fourier analysis. For ex-

ample:

Theorem 3.1 (Diaconis, [24]). For G a group, Q a probability distribution on G,

and U the uniform distribution on G,

|Q− U |2TV ≤1

4

∑ρ

dρ Tr(Q(ρ)Q(ρ)∗),

where ∗ denotes conjugate transpose and the sum is over all nontrivial irreducible

representations ρ of G.

Diaconis and Ram find similar such bounds in their analysis of the Metropolis

algorithm applied to the symmetric group [25]. Both Theorem 3.1 and the results

of [25] require the notion of Fourier inversion and and Plancherel’s Theorem (see

Theorem 3.2 below).

Definition 3.6. For A a semisimple algebra, a trace function on A is a C-linear

109


function τ : A→ C such that for all a, b ∈ A,

τ(ab) = τ(ba).

Note that by linearity the usual trace function on Md(C) is in fact unique up to

multiplication by a constant. Hence, for any trace τ on A and set R of inequivalent

irreducible representations of A, there exist constants tρ ∈ C such that:

τ =∑ρ∈R

tρTρ,

where for a ∈ A, Tρ(a) = Tr(ρ(a)).

A trace function τ gives rise to a symmetric bilinear form 〈·, ·〉τ : A× A → C by

letting

〈a, b〉τ = τ(ab),

for a, b ∈ A.

Theorem 3.2 (Diaconis, Ram [25]). Let A be a semisimple algebra with basis {ai}

and τ a nondegenerate trace on A. Let {a∗i } be the dual basis to {ai} with respect to

the trace form 〈·, ·〉τ . Then for f, f1, f2 complex-valued functions on A,

f(ai) =∑ρ

tρ Tr(f(ρ)ρ(a∗i )), (3.1)

〈f1, f2〉τ =∑ρ

tρ Tr(f1(ρ)f2(ρ)). (3.2)

110


3.3.2 The Brauer Algebra

Recall from Chapter 2, Section 2.5.1, that an element in the symmetric group Sn

can be realized as a diagram on 2n points, consisting of two rows of n points each,

with each point in the top row connected by an edge to one in the bottom row.

Multiplication in the symmetric group is realized as concatenation of diagrams. The

simple transpositions {ri = (i i + 1) | 1 ≤ i ≤ n − 1} form a generating set for the

symmetric group, allowing us to define a natural length function lS : Sn → N on the

symmetric group: for w ∈ Sn, let lS(w) be the minimum number of generators needed

to express w.

Elements of the Brauer monoid, Brn, are realized as generalized symmetric group

diagrams: consider diagrams on 2 rows of n points each, with edges connecting pairs

of points regardless of row and each point part of exactly one edge. Multiplication

may again be realized as concatenation of diagrams. Note that in some cases, this

introduces a closed loop. A parameter q is used to keep track of the number of closed

loops: for two diagrams x, y ∈ Brn, let c denote the number of closed loops in the

multiplication xy and let z be the diagram of this product with the closed loops

removed. Then xy = qcz.

=x

yq z

Figure 3.1: xy = q1z

Two Brauer diagrams d1 and d2 are equivalent if they differ only in the number

of closed loops, i.e., if when q = 1, d1 = d2. For example, for x, y, z as in Figure

111


3.1, the product xy is equivalent to z. The Brauer monoid, Brn consists of the set

of equivalence classes of such diagrams and is generated by {ri, ei | 1 ≤ i ≤ n − 1}

(see Figure 2.40). The symmetric group Sn, generated by the transpositions {ri | 1 ≤

i ≤ n− 1}, sits inside of Brn. As in the symmetric group, a natural length function

lBr : Brn −→ N exists for the Brauer monoid: for w ∈ Brn, define lBr(w) to be the

minimum number of generators ({ri, ei}) needed to express w.

. . . . . .

i i+ 1

ri

. . . . . .

i i+ 1

ei

Figure 3.2: ri, ei ∈ Brn

The Brauer algebra, Brn, is the C(q)-algebra with basis Brn. Equivalently (see,

for example [5]), Brn has algebraic presentation given by generating set

{ri, ei | 1 ≤ i ≤ n− 1},

along with relations:

(1) r2i = 1, (2) rirj = rjri, riej = ejri, eiej = ejei, |i− j| > 1

(3) e2i = qei, (4) eiri = riei = ei,

(5) riri+1ri = ri+1riri+1, (6) eiei+1ei = ei, ei+1eiei+1 = ei+1,

(7) riei+1ei = ri+1ei, (8) ei+1eiri+1 = ei+1ri.

112


3.3.3 The BMW Algebra

Elements of the BMW monoid are realized as generalized Brauer diagrams called

tangles. A tangle is again a diagram on 2 rows of n points each with edges connecting

pairs of points regardless of row with each point part of exactly one edge. At each

crossing of two edges we distinguish which edge passes above and which passes below

(see Figure 3.3). As in the Brauer monoid, multiplication is concatenation of diagrams

and two tangles are equivalent if they differ only in their number of closed loops.

Figure 3.3: A Tangle

Further, two tangles are equivalent if they are related by a sequence of Reidemeis-

ter moves of type II and III:

RII :

RIII :

←→

←→

Consider the elements Tri , T−1ri

, and Tei of Figure 3.4.

. . . . . .

i i+ 1

Tri

. . . . . .

i i+ 1

Tei

113


. . . . . .

i i+ 1

T−1ri

Figure 3.4: Tri , Tei , T−1ri

A tangle is reachable if it can be obtained as a finite product of elements from

{Tri , Tei , T−1ri| 1 ≤ i ≤ n − 1}. The BMW monoid, BMWn, consists of the set of

equivalence classes of reachable tangles on 2n points.

For m, `, q parameters satisfying q = (`−`−1)(m−m−1)−1 +1, the BMW algebra,

BMWn, is the C(q,m, `)-algebra with basis BMWn and the following untangling

relations:

= +m −m

Equivalently (see, for example [40]), the BMW algebra has algebraic presentation

114


given by generating set {Tei , Tri , T−1ri| 1 ≤ i ≤ n− 1}, along with relations:

(A1) T 2ei

= qTei , (A2) TeiTri = TriTei = `−1Tei

(A3) TeiTei±1Tei = Tei , (A4) TeiTri±1

Tei = `Tei ,

(A5) TriTri+1Tri = Tri+1

TriTri+1, (A6) TriTri±1

Tei = Tei±1Tei = Tei±1

TriTri±1,

(A7) Tri = T−1ri

+mTid −mTei (A8) TriTrj = TrjTri , TriTej = TejTri ,

TeiTej = TejTei , |i− j| > 1,

for q = (`− `−1)(m−m−1)−1 + 1 and Tid the identity element.

We map an element of the BMW monoid to the Brauer monoid by ‘forgetting’

the crossing information. Denote this map by φ : BMWn −→ Brn.

Example 3.1. For x the tangle of Figure 3.3, φ(x) has form:

Further, each element of the Brauer monoid lifts to the BMW algebra: for d ∈ Brn,

the BMW image of d, Td, realizes d as a tangle by redrawing the edges of d from right

to left across the first dn2e points in the bottom row, lifting the pen when crossing an

edge that has already been drawn, then moving to the top row of points and drawing

all horizontal edges in this row, again lifting the pen when crossing an edge that has

already been drawn, and finally drawing the remaining edges of d from right to left

across the bottom row of points.

Example 3.2. For d the Brauer diagram of Example 3.1, the BMW image of d is:

115


Note that when ` = 1 the BMW image of d has a simple algebraic description: for

d ∈ Brn and si ∈ {ri, ei}, define a reduced expression for d to be an expression d =

si1si2 · · · sik for d that has each occurrence of ei±1ri replaced by ei±1eiri±1 (whenever

this occurrence does not respect the method of drawing above) and has minimal

length over all such expressions for d.

Then the BMW image of d, Td, realizes d as a tangle by setting

Td := Tsi1Tsi2 · · ·Tsik ,

for d = si1si2 · · · sik a reduced expression that contains the maximum number of e

terms over all reduced expressions of d. This allows us to define a natural length

function L : Tn −→ N as

L(Td) = l′Br(d) + maxs

(e(s))

over all reduced expressions s of d, where e(s) gives the number of e terms in s and

l′Br(d) gives the minimum number of generators needed for a reduced expression of d.

Example 3.3. Let d = r3e2e1r3, so

Td = Tr3Te2Te1Tr3 ,

and L(Td) = l′Br(d) + 2 = 6. An alternate reduced expression for d with the same

number of e terms is d = r3e2r3e1, which has the same BMW image by BMW relation

(A8):

Tr3Te2Tr3Te1 = Tr3Te2Te1Tr3 .

116


An additional reduced expression for d with the same number of e terms is d =

r2e3r2e1, but we must replace e3r2 using Brauer relation (8) as follows:

d = r2e3r2e1 = r2e3e2r3e1,

but then using Brauer relation (7),

d = r3e2r3e1,

as before.

Theorem 3.12 of [45] shows that the BMW images of the Brauer monoid elements

form a basis for BMWn. We denote this basis by Tn := {Td | d ∈ Brn}.

Define functions Tri ,Tei : Tn −→ BMWn as follows: for x ∈ Tn,

Tri(x) = Trix

Tei(x) =

Teix if Trix /∈ Tn,

Trix else.

We consider generation of elements in Tn via random walks on Tn and translate

these walks into left multiplication by Tri and Tei .

117

3.4 The Random Walk

3.4 The Random Walk

In the finite group case, left multiplication by a generating set gives rise to a random

walk on the group. For example, for each generator ri of Sn, consider the probability

distribution

Pi(x, y) =

1 if y = rix,

0 else.

Then setting π to be the probability distribution

π(x) =θ−lS(x)∑

w∈Sn

θ−lS(w),

the Metropolis algorithm construction yields

Mi(x, y) =

1 if y = rix and lS(y) > lS(x),

θ if y = rix and lS(y) < lS(x),

1− θ if y = x,

where lS represents the length function on words in Sn.

Interpreted as a random walk on Sn, the chain Mi describes the process:

From x ∈ Sn multiply by ri. If the length increases, move to

rix. If the length decreases, flip a θ-coin and if heads move

to rix. If tails, remain at x.

(∗)

We generalize this walk to the basis of tangles Tn of the BMW algebra. For the

remainder of this chapter, let ` = 1 in the BMW algebra. For Td ∈ Tn and L the

118

3.4 The Random Walk

length function on Tn defined in Section 3.2, let

π(Td) =θ−L(Td)∑

w∈Tn

θ−L(w),

and for y ∈ Tn let

P ′i (Td, y) =

1 y = Trid

0 else.

Then the Metropolis algorithm applied to P ′ with probability distribution π yields:

Ki(Td, y) =

1 if y = Trid and L(y) ≥ L(Td),

θ if y = Trid and L(y) < L(Td),

1− θ if y = Td.

Remark 3.1. Recall that Sn ⊆ Brn and note that for d ∈ Sn, L(Td) = l′Br(d) =

lBr(d) = lS(d), where L, l′Br, lBr, and lS denote the length functions on Tn, Brn, and

Sn as defined in Section 3.2. Thus, the submatrix of Ki corresponding to states

{Td | d ∈ Sn} is exactly the chain Mi of [25].

Interpreted as a random walk on Tn, the chain Ki describes the process:

From Td ∈ Tn consider d ∈ Brn and multiply by ri. If the length

of the BMW image, Trid, increases, move to it. If the length decreases,

flip a θ-coin and if heads move to Trid. If tails, remain at Td.

(†)

119

3.4 The Random Walk

In light of Proposition 3.1 below, Walk (†) can be rephrased as:

From Td ∈ Tn multiply by Tri . If the result is an element of Tn,

move to TriTd. Else, flip a θ-coin and if heads move to T−1riTd.

If tails, remain at Td.

(††)

Rephrasing in this way yields the equivalent corresponding Markov chain:

Ki(x, y) =

1 if y = Trix,

θ if y = T−1rix,

1− θ if y = x.

An example of Ki can be found in Appendix B.1.

Proposition 3.1. For Td ∈ Tn,

L(Trid) < L(Td) ⇐⇒ TriTd /∈ Tn.

Further, if TriTd /∈ Tn, then T−1riTd = Trid ∈ Tn, while if TriTd ∈ Tn, then TriTd = Trid.

Proof. First write Td = Tsi1Tsi2 · · ·Tsik , for si1 · · · sik a reduced expression for d with

maximum number of e terms. Then

TriTd = TriTsi1Tsi2 · · ·Tsik ,

which, after possibly rearranging using BMW relations (A5) and (A8), has one of the

following forms, for some 1 ≤ j ≤ k − 2:

(a) TriTd = Tsi1Tsi2 · · ·TsijTriTsiTsi±1Tsij+3

· · ·Tsik

120

3.4 The Random Walk

(b) TriTd = Tsi1Tsi2 · · ·TsijTriTsi±1TsiTsij+3

· · ·Tsik ,

(c) TriTd = TriTsi1Tsi2 · · ·Tsik , |i1 − i| > 1.

The proof then reduces to checking each possible case. For example, if in case (1)

with

TriTd = Tsi1Tsi2 · · ·TsijTriTriTei+1Tsij+3

· · ·Tsik ,

then since TriTri /∈ Tn, we see that TriTd /∈ Tn. Further, since BMW relations (A5)

and (A8) hold in the Brauer monoid,

rid = risi1 · · · sik = si1si2 · · · sijririei+1sij+3· · · sik ,

which by Brauer relation (1) gives

rid = si1 · · · sijei+1sij+3· · · sik ,

and so l′Br(rid) = l′Br(d)− 1 = k − 1.

Now let d′ = rid and recall for w ∈ Brn, e(w) gives the maximum number of

e terms over all reduced expressions for w. If e(d′) > e(d), then there is an ex-

pression for d′ with length k − 1 and at least one more additional e term than in

si1 · · · sijei+1sij+3· · · sik . However, by Brauer relations (1)-(8), the only way to in-

crease the number of e terms without changing the length of an expression is through

the relation

ei+1riri+2ei+1 = ei+1eiei+2ei+1.

This would require either sij−2sij−1

sij = ei+1riri+2 or sij+3sij+4

sij+5= riri+2ei+1,

which contradicts the fact that si1 · · · sik is a reduced expression for d with maximal

121

3.4 The Random Walk

number of e terms. Hence, e(d′) ≤ e(d), and so

L(Trid) < L(d).

For the second statement, note that

T−1riTd = Tsi1Tsi2 · · ·TsijT

−1riTriTei±1

Tsij+3· · ·Tsik = Trid.

The remaining cases are checked similarly.

In [25], Diaconis and Ram translate the Markov chain Mi into left multiplication

by Hecke algebra elements on a suitably chosen basis. Similarly, we translate the

chains Ki arising from our Metropolis construction into left multiplication by BMW

algebra elements on the basis Tn. Recall the functions Tri ,Tei : Tn −→ Tn defined at

the end of Section 3.2.

Theorem 3.3. Let Brn be the Brauer monoid and BMWn the BMW algebra with

basis Tn = {Td | d ∈ Brn} as defined in Section 3.2. Let m = (1− θ)(θ)−1 and ` = 1.

Then the chain Ki is the same as the matrix of left multiplication by

θTri + (1− θ)Tei ,

with respect to the basis Tn of BMWn.

Proof. Let x ∈ Tn and consider left multiplication by Tri . If Trix ∈ Tn,

(θTri + (1− θ)Tei)x = θTrix+ (1− θ)Trix = Trix.

122

3.5 Analysis of the Walk

If Trix /∈ mathcalTn then by BMW Relation (A7),

Trix = (T−1ri

+mTid −ml−1Tei)x = T−1rix+ (1− θ)(θ)−1x− (1− θ)(θ)−1Teix.

By Proposition 3.1, T−1rix ∈ Tn, and

(θTri + (1− θ)Tei)x = θT−1rix+ (1− θ)x− (1− θ)Teix+ (1− θ)Teix.

= θT−1rix+ (1− θ)x,

The chains Ki provide scanning strategies for generating elements of the BMW

and Brauer monoids:

1

n− 1

n−1∑i=1

Ki (random scan),

K1K2 · · ·Kn−1Kn−1 · · ·K2K1 (short systematic scan),

(K1 · · ·Kn−1Kn−1 · · ·K1) · · · (K1K2K2K1)(K1K1) (long systematic scan).

Theorem 3.3, coupled with the results of Section 3.5, allows for the study of the rate

of convergence of the systematic scans arising from the chains Ki in terms of the

representation theory and Fourier analysis of the BMW algebra.


We first discuss the stationary distribution of the random scan, short systematic scan,

and long systematic scan described in Section 3.4. For the remainder of the section,

we use K to refer to the matrix corresponding to any of these scans, as the results

123


hold true for all three scans.

Note that K is Markov and recall that a communication class C of a Markov

chain is closed if for each state x ∈ C and for all y /∈ C, y is not accessible from

x. We determine the closed communication classes of K and analyze the stationary

distribution of each closed communication class.

The communication classes of K depend on the number of lower horizontal edges

in the tangle diagrams for the states.

Definition 3.7. Let x ∈ Tn. An edge of x is lower (respectively, upper) hori-

zontal if it connects two points that are both on the bottom (respectively, top) row of

the diagram of x.

Example 3.4. In Figure 3.5, E3 is the only lower horizontal edge and E1 is the only

upper horizontal edge.

E1

E2

E4E3

Figure 3.5

K is determined by left multiplication by Tri and T−1ri

only, and left multiplication

by these elements cannot affect existing lower horizontal edges in a tangle diagram, nor

can it create new ones. Thus, the communication classes of K consist of those states

with the same lower horizontal edges. For xi ∈ Tn, let Xi denote its communication

class:

Xi := {y ∈ Tn | lower horizontal edges of y the same as those of xi}.

124


For each communication class Xi, let [K]i denote the corresponding submatrix of

K. Note that the communication class for x0 := Tid consists of the states {Td | d ∈

Sn}. Then by Remark 3.1, [Ki]0 = Mi, and so [K]0 can be analyzed using the methods

of [25]. For the remainder of the chapter we consider the remaining communication

classes of K:

{Xi | i 6= 0}.

To analyze the time to stationarity of the submatrix [K]1 corresponding to a

communication class Xi, we pair X1 with another communication class, X2, whose

states have the same number of lower horizontal edges as those in X1. For w ∈ X1,

let w∗ denote the element of X2 with the same upper configuration as w. Define the

matrix:

K(x, y) =

K(x, y) if x, y ∈ X1 or if x = w∗, y = z∗ for w, z ∈ X1,

1 if x = y, x /∈ X1 ∪X2,

0 else.

Note with Example 3.5 that often K(x, y) = K(x∗, y∗).

Example 3.5. For T3, let x1 = Te1 and x2 = Te1Tr2 , so X1 = {Te1 , Tr2Te1 , Te2Te1},

X2 = {Te1Tr2 , Tr2Te1Tr2 , Te2Te1Tr2}, T ∗e1 = Te1Tr2 , Tr2T∗e1

= Tr2Te1Tr2 , and Te2T∗e1

=

Te2Te1Tr2 .

125


Then for K = 12(K1 +K2),

2[K]1 =

Te1 Tr2Te1 Te2Te1

1 θ 0

1 1− θ θ

0 1 2− θ

,

2[K]2 =

Te1Tr2 Tr2Te1Tr2 Te2Te1Tr2

1 θ 0

1 1− θ θ

0 1 2− θ

,

and K = [K]1⊕

[K]2⊕

I9, for I9 the 9× 9 identity matrix.

Let π denote the stationary distribution of K. For Tx ∈ Tn let [π]x denote the

column of π corresponding to Tx:

[π]x :=

πx(x1)

πx(x2)

...

πx(xk)

=∑Ty∈Tn

πx(y)Ty.

Note that πx(y) represents the probability of ending at state Ty after starting at Tx.

Recall from Section 3.2.1 that to analyze the time to stationarity of K we consider:

|Kmx − π|TV . (3.3)

126


By Lemma 3.1, the total variation norm is bounded by the L2(π)-norm, so we

translate the L2(π)-norm into a trace norm on BMWn in order to use the represen-

tation theory of the BMW algebra to analyze (3.3).

Definition 3.8. Define τ : Tn → C as follows: for x ∈ Tn,

τ(x) =

1 if x = Tid,

0 else,

The restricted trace, τ : BMWn → C, is the linear extension of τ to BMWn.

Recall from Section 3.2 that Tn = {Td | d ∈ Brn}.

Proposition 3.2. For Tx, Ty ∈ Tn, τ(TxTy) =

1 if x = y−1,

0 else.

Corollary 3.1. τ is a trace function on BMWn.

Proof of Proposition 3.2. Let Tx, Ty ∈ Tn. By definition, Tx = Tsi1 · · ·Tsik , where for

each 1 ≤ j ≤ k, Tsij ∈ {Tri , Tei | 1 ≤ i ≤ n − 1}. First note that the LHS of any of

the BMW relations (A1)-(A8) has a Te factor if and only if the RHS also has at least

one Te factor. In particular, if for some 1 ≤ j ≤ n− 1, Tej is a factor of Tx, then each

term of the product TxTy has at least one Te factor. Hence, no term in the product

TxTy is the identity, so τ(TxTy) = 0. Similarly, τ(TyTx) = 0.

Thus, if Tsil = Tej for some 1 ≤ l ≤ k, 1 ≤ j ≤ n− 1, then τ(TxTy) = τ(TyTx) = 0

for all Ty ∈ Tn. Equivalently, τ(TxTy) = 0 for all x ∈ Brn − Sn, y ∈ Brn.

Next note that Tx ∈ Tn has an inverse iff x ∈ Sn ⊂ Brn, because the diagram

for any y ∈ Brn − Sn has at least one lower horizontal edge, and left multiplication

127


cannot affect this edge. Hence we need show for x, y ∈ Sn that

τ(TxTy) =

1 if x = y−1,

0 else.

As in Section 3.2 let lS : Sn → N denote the length function of Sn. We proceed

by induction on the length lS(x) + lS(y).

If lS(x) + lS(y) = 0 then lS(x) = lS(y) = 0, so x = y = id. Clearly τ(TxTy) = 1

and x = y−1.

Now suppose the result holds true for all w1, w2 ∈ Sn with lS(w1)+ lS(w2) ≤ n−1.

Let x, y ∈ Sn with ri1 · · · rik a reduced expression for x, rj1 · · · rjl a reduced expression

for y, and k + l = n. Then Tx = Tri1 · · ·Trik , Ty = Trj1 · · ·Trjl , and

TxTy = Tri1 · · ·TrikTrj1 · · ·Trjl .

If TxTy ∈ Tn then no matter the rearrangement using BMW Relations (A5) and

(A8), T 2ri

must not occur for any i. Then r2i does not occur in xy, so lS(xy) = n. In

particular, x 6= y−1 and Tid 6= TxTy = Txy ∈ Tn, so τ(TxTy) = 0.

Suppose now that TxTy /∈ Tn. Then, after applying Relations (A5) and (A8) and

possibly reindexing, the product TxTy has form:

TxTy = (Tri1 · · ·Trik )(TrikTrj2 · · ·Trjl ) = Tx′TrikTrikTy′ ,

128


for x′ = ri1 · · · rik−1and y′ = rj2 · · · rjl . Then by BMW relation (A7),

TxTy = Tx′(Tid −mTrik +ml−1Teik )Ty′

= Tx′Ty′ −mTx′TrikTy′ +ml−1Tx′TeikTy′ .

(3.4)

Since Tx′TeikTy′ ∈ Brn − Sn, τ( ˜Tx′TeikTy′) = 0.

Now consider τ(Tx′TrikTy′) = τ(TxTy′). Since lS(x) + lS(y′) = n − 1, by the

induction hypothesis,

τ(TxTy′) = 1 ⇐⇒ y′ = x−1,

and is zero otherwise.

But if y′ = x−1, then lS(y′) = lS(x) = k and y′ = rik · · · ri1 . Since lS(y′) = lS(y)−1,

this implies that

lS(y) = k + 1.

However, y = riky′ = rikrik · · · ri1 , and so

lS(y) = k − 1.

With this contradiction, we have τ(TxTy′) = 0.

Finally, consider τ(Tx′Ty′). Since lS(x′) + lS(y′) = n − 2, then by the induction

129


hypothesis,

τ(Tx′Ty′) = 1

⇐⇒ x′ = (y′)−1

⇐⇒ y′ = rik−1· · · ri1

⇐⇒ y = rikrik−1· · · ri1

⇐⇒ y = x−1,

(3.5)

and τ(Tx′Ty′) = 0 otherwise.

Thus,

τ(TxTy) =

1 if x = y−1,

0 else.

Thus τ is a trace function on BMWn with τ(TxTy) = 0 for all x, y ∈ Brn − Sn.

In fact, τ extends the natural trace function of the Hecke algebra, Hn, viewing Hn as

a subalgebra of BMWn. We analyze K using the bilinear form arising from τ , which

reformulates questions about the time to stationarity in terms of the representation

theory of the underlying Hecke subalgebra of BMWn.

Recall that K is comprised of two submatrices corresponding to two communica-

tion classes X1 and X2 of K. Recall further that these communication classes consist

of elements of Tn with at least one lower horizontal edge. In particular, for each

Tx ∈ X1 ∪ X2, τ(TxTy) = 0 for all Ty ∈ Tn. In order for τ to be nontrivial on the

communication classes of K, we rewrite K with respect to a shifted basis for BMWn.

Definition 3.9. Let π denote the stationary distribution of K. To each Tx ∈ X1,

130


associate a distinct sx ∈ Sn such that sx 6= s−1y for all Ty ∈ X1 and sx has order

greater than 2. For Tx ∈ X1 and for Ty /∈ X1 ∪X2, let

Tx := Tx + πx(x)−12Tsx ,

Tx∗ := Tx∗ + πx(x)−12Tsx−1 = Tx∗ + πx∗(x

∗)−12Tsx−1 ,

Ty := Ty.

(3.6)

We note in Appendix B.2 that for all communication classes corresponding to ele-

ments with at least two lower horizontal edges, Sn contains enough distinct elements

to make the associations of Definition 3.9. For the remainder of this section let X1

be a communication class whose elements contain at least two lower horizontal edges.

(The remaining communication classes are analyzed separately through techniques

discussed in Appendix B.2.)

Lemma 3.2. Tn := {Tx | x ∈ Brn} is a basis for BMWn.

Proof. This follows from the fact that Tn is a basis for BMWn.

Now let 〈 , 〉BMW denote the trace form of Section 3.3.1

Lemma 3.3. For Tx ∈ X1 ∪X2 and y ∈ Brn,

〈Tx, Ty〉BMW =

πx(x)−1 if y = x∗,

πx(x)−12 if y = (sx)

−1,

0 else,

131


while

〈Tsx , Ty〉BMW =

πx(x)−

12 if y = x∗,

1 if y = (sx)−1,

0 else.

Proof. Follows from Proposition 3.2 and the linearity of trace.

Let K be the matrix of K with respect to Tn. Note that time to stationarity is

invariant under change of basis.

Lemma 3.4. For Tx ∈ Tn,

(a) If Tx ∈ X1,

K(Tx, Ty) =

K(Tx, Ty) if Ty ∈ X1 ∪ X2,

(1−K(Tx, Tx))πx(x)−12 if y = sx,

−K(Tx, Ty)πy(y)−12 if y = sz, z 6= x, z ∈ X1

0 else,

and similarly for Tx ∈ X2.

(b) If Tx /∈ X1 ∪ X2,

K(Tx, Ty) =

1 if y = x,

0 else.

Proof. Follows from definition of K and Tn.

Recall that K is the direct sum K1

⊕K2

⊕Im, where K1 corresponds to the

elements in X1, K2 to those in X2, and m = |Tn| − 2|X1|. Lemma 3.4 shows that K

is also a direct sum K1

⊕K2

⊕Im, where for i = 1, 2, the matrix Ki corresponds to

132


{Tx, Tsx | Tx ∈ Xi}, and m = |Tn| − 4|X1|. Further, for all Tx, Ty ∈ X1 ∪X2,

K(Tx, Ty) = K(Tx, Ty) = K(Tx, Ty).

Recall that π denotes the stationary distribution of K and [π]x the stationary

distribution of K corresponding to column Tx:

[π]x =∑Ty∈Tn

πx(y)Ty.

Note that for Tx /∈ X1 ∪X2, πx(y) = 0 for all Ty 6= Tx, and so

[π]x = Tx.

Further, for Tx ∈ X1, πx(y) = 0 for all Ty /∈ X1, and so

[π]x =∑Ty∈X1

πx(y)Ty,

and similarly for X2. Finally, note that πx(y) = πx′(y) for all x, x′, y ∈ X1, and

similarly for X2.

Let π denote the stationary distribution of K and [π]x the stationary distribution

of K corresponding to column Tx. Let Xi = {Tx | Tx ∈ Xi}.

Lemma 3.5. Let π be the stationary distribution of K and π the stationary distri-

bution of K.

133


(a) For Tx ∈ X1,

[π]x =∑Ty∈X1

(πx(y)Ty − πx(y)12 Tsy) + πx(x)−

12 Tsx ,

and similarly for Tx ∈ X2.

(b) If Ty /∈ X1 ∪ X2, [π]y = Ty

Proof. Part (2) follows from Lemma 3.4. To prove (1), note that for Tx ∈ X1

[π]x = [π]x + πx(x)−12 [π]sx

=∑Ty∈X1

πx(y)Ty + πx(x)−12Tsx

=∑Ty∈X1

(πx(y)(Ty + πy(y)−

12Tsy)− πx(y)πy(y)−

12Tsy

)+ πx(x)−

12Tsx .

=∑Ty∈X1

(πx(y)Ty − πx(y)

12 Tsy

)+ πx(x)−

12 Tsx .

For Tx ∈ X1 ∪ X2, Lemma 3.5 shows that πx(y) = πx(y) for all Ty ∈ X1 ∪ X2.

However, πx(sy) = 0, but πx(sx) = πx(x)−12 − πx(x)

12 and πx(sy) = −πx(y)

12 , for

y 6= x.

Let S := {Tsx | sx ∈ S}. Consider the L2(π)-norm restricted to the subspace

generated by X1 ∪ X2 ∪ S:

Definition 3.10. For functions f, g : X1 ∪ X2 ∪ S → C, let

〈f, g〉2 :=∑

Tx∈X1∪X2∪S

f(x)g(x)πx(x).

134


For m ∈ N, let [Km]x denote the column of Km corresponding to Tx:

[Kn]x =∑

Txi∈X1∪X2∪S

Knx (xi)Txi .

To find the time to stationarity of K (and hence K andK), we analyze ‖[Km]x−[π]x‖2.

Lemma 3.6. Let f, g be complex-valued functions on X1 ∪ X2 ∪ S and let ∗ : X1 ∪

X2 ∪ S → X1 ∪ X2 ∪ S be the involution that sends Tx to Tx∗ for Tx ∈ X1 ∪ X2, and

Tsx to Ts−1x

for Tsx ∈ S. Then for 〈 , 〉BMW the bilinear form arising from the trace

τ ,

〈f/π, g/π〉2 = 〈f, g∗〉BMW −∑Tsx

f(x)g(s−1x ) + f(s−1

x )g(x∗)

πx(x)12

.

Proof. By Lemma 3.3,

〈f/π, g/π〉2 =∑ f(x)g(x)

πx(x)

=∑

Tx∈X1∪X2∪S

f(x)g(x)〈Tx, (Tx)∗〉BMW

=∑

Tx,Ty∈X1∪X2∪S

f(x)g(y)〈Tx, (Ty)∗〉BMW −∑Tsx∈S

f(x)g(s−1x )

πx(x)12

−∑Tsx∈S

f(s−1x )g(x∗)

πx(x)12

= 〈f, g∗〉BMW −∑Tsx

f(x)g(s−1x ) + f(s−1

x )g(x∗)

πx(x)12

.

135


Corollary 3.2. For Tx ∈ X1,

〈[Km/π]x, [Km/π]x〉2 = 〈[Km]x, [K

m]x〉BMW −∑Ty∈X2

Kmx (s−1

y )Kmx (y∗)

πy(y)12

.

Proof. Follows from Lemma 3.6 and the fact that ([Km]x)∗ = [Km]x, K

mx (s−1

y ) = 0 if

y ∈ X1, and Kmx (y) = 0 if y ∈ X2.

K is Markov, so there exists N ∈ N with Kmx ≥ 0 for all m > N . Further, π is

the stationary distribution of a Markov chain, so πy(y) ≥ 0. We have thus shown:

Theorem 3.4. For Tx ∈ X1 ∪ X2,

〈[Kn/π]x, [Kn/π]x〉2 ≤ 〈[Kn]x, [K

n]x〉BMW .

Hence,

‖[Kn/π]x − 1‖22 ≤ ‖[Kn]x − 1‖2

BMW .

Thus, studying the time to stationarity of K can be achieved by studying

‖[Kn]x − 1‖2BMW .

136

Chapter 4

Future Directions

4.1 Further Directions

Our work exploits the interplay between algebra, combinatorics, and probability the-

ory, and we look forward to extending this research program by continuing to explore

such links. In particular, we wish to consider the following ideas:

• Extension of the results of the SOV approach to other chains of sub-

algebras for the Weyl groups, the general linear groups of finite fields,

and the Brauer algebra.

The efficiency of the SOV approach of Chapter 2 in part depends on the choice of

subgroup (subalgebra) chain. While we provide efficiency counts that improve

upon previous bounds [63, 65], we do not consider maximally dense subgroup

(subalgebra) chains. We wish to investigate the change in efficiency counts upon

considering alternate chains. For example, for the Brauer algebra we wish to

consider the algebra chain:

137


Brn > Br2 × Brn−2 > Brn−2 > · · · > Br0.

• Extension of the results of the SOV approach to other semisimple

algebras, and applications of such transforms.

While our algorithm in Chapter 2 allows for the computation of generalized

Fourier transforms by factorization through a subgroup (subalgebra) chain, the

efficiency of the algorithm depends on the choice of this subgroup (subalgebra)

chain as well as knowledge of the irreducible representations and associated

Bratteli diagram. There are many particular semisimple algebras to explore in

this manner. In [43], Grood constructs the irreducible representations of the

rook partition algebra and the associated Bratteli diagram, while Halverson et

al. [35, 44] determine analogues of the seminormal representations of Sn for the

rook-Brauer algebra and planar-rook algebra. Motivated by these results, we

plan to extend our algorithm to these algebras by developing an understanding

of the centralizers and irreducible representations and to explore the resulting

combinatorial path-counting questions to provide efficient counts.

• Further analysis of the random walk on the BMW algebra and ex-

tension to other semisimple algebras.

In Chapter 3 we develop a systematic scanning strategy for generating elements

of the BMW monoid basis of the BMW algebra, then view the associated matrix

as a left multiplication operator in the BMW algebra. We further develop

a trace norm on the BMW algebra and prove that bounds on the time to

stationarity of the walk can be determined by bounding this trace norm. Using

138


similar techniques in the Coxeter group setting, Diaconis and Ram [25] provide

explicit bounds determined by the dimensions and traces of the irreducible

representations of the Hecke algebra. We plan to exploit the representation

theory of the BMW algebra to provide explicit bounds in terms of the traces of

irreducible representations of the BMW algebra. We wish also to extend these

ideas to other diagram algebras such as the rook partition algebra, planar-rook

algebra, and rook-Brauer algebra.

• Extension of generalized Fourier transforms to buildings and appli-

cation to random walks on p-adic groups.

We would like to extend our work on Fourier transforms to the setting of Bruhat-

Tits buildings. Motivated by an interest in random walks on buildings, which

begins with Sawyer’s treatment of the binary tree Gl2(Q2) [84] and continues

with the work of Cartwright et al. [12] and Parkinson [72], we wish to determine

the appropriate generalization of the Fourier transform in the buildings context

in order to apply Fourier theory to questions of random walks on buildings and

p-adic groups.

139

Appendix A

Appendix

A.1 Gel’fand-Tsetlin Bases and Adapted Repre-

sentations

Definition A.1. Given a Bratteli diagram B, a representation of B assigns to each

α ∈ V (B) a linear space Vα and to each edge e ∈ E(B) a linear map Te : Ve− → Ve+.

Given two representations ({Vα}α∈V (B), {Te}e∈E(B)), ({Wα}α∈V (B), {Se}e∈E(B)), of B,

a morphism m : V → W is a family of linear maps {mα : Vα → Wα}α∈V (B) such

that the diagram

Ve−Te−−→ Ve+

me− ↓ ↓ me+

We−Se−−→ We+

commutes for all e ∈ E(B).

A model representation of B is a representation of B such that for all e ∈

140

A.1 Gel’fand-Tsetlin Bases and Adapted Representations

E(B), Te is injective, and for all nonroot vertices α ∈ V (B),

Vα =⊕e+=α

Im(Te).

Definition A.2. Given a chain of subalgebras {Ai}, a model representation for

{Ai} is a model representation of the corresponding Bratteli diagram B such that:

(i) for each α ∈ V (B) at level i, Vα is the representation space of the representation

of Ai corresponding to α,

(ii) for each e ∈ E(B) from level i to level i + 1, Te is Ai-equivariant, i.e., for ρe−

the representation of Ai corresponding to e− and ρe+ the representation of Ai+1

corresponding to e+, the diagram


ρe− ↓ ↓ ρe+


commutes for all e ∈ E(B).

A model representation of an algebra chain has a natural basis of paths:

Lemma A.1. Given a model representation of a chain of subalgebras with Bratteli

diagram B, the collection of distinct paths in B from the root to a vertex α ∈ V (B)

corresponds to a choice of basis for Vα.

Proof. Consider the space Vβ corresponding to the root β, i.e., Vβ is the representation

space of C[B0] = C, so Vβ is one-dimensional. Now let α be a vertex in B with

141

A.1 Gel’fand-Tsetlin Bases and Adapted Representations

gr(α) = 1. Then Vα =⊕e+=α

Im(Te) ∼=⊕e+=α

Vβ since Te is injective. Induction gives

the result.

Thus, given an irreducible representation ρ of Ai corresponding to a vertex α in

the Bratteli diagram associated to the chain of subalgebras {Ai}, there is a basis for

the representation space of ρ indexed by the paths from the root to α. We call such

a basis a Gel’fand-Tsetlin basis, as in Definition 2.10. Given a model representation

of a Bratteli diagram B, Lemma A.1 gives a system of Gel’fand-Tsetlin bases for B.

In fact, these are equivalent concepts:

Theorem A.1. A system of Gel’fand-Tsetlin bases for a Bratteli diagram B uniquely

determines a model representation for B. Conversely, a model representation uniquely

determines a system of Gel’fand-Tsetlin bases for B.

Proof. Both require a choice of vector space for each vertex of B, so we need only

show how a choice of basis corresponds with linear maps Te for each edge e.

Given a system of bases and an edge e ∈ B, a basis vector for Ve− corresponds

to a path P from the root to e−. Then e ◦ P is a path from the root to e+, which

corresponds to a basis vector for Ve+ . In other words, we have an injection of Ve− into

Ve+ .

Conversely, given a model representation and a vertex α, every path from the root

to α corresponds to an injection of C into Vα. Since

Vα =⊕e+=α

Im(Te),

the union of the distinct images of 1 ∈ C over the collection of injections gives a basis

for Vα as we vary over all possible paths from the root to α.

142

A.2 Restricted Product Lemmas

Remark A.1. The equivalent definitions of Gel’fand-Tsetlin bases and model repre-

sentations coincide with the notion of a complete set of adapted representations for

chains of groups. Clearly a model representation for the group algebra chain gives

rise to an adapted basis since the isomorphism

Vα =⊕e+=α

Im(Te) ∼=⊕

e∈E(B),e+=α

Ve−

describes how the representation space Vα decomposes at level i− 1. Equivariance of

the maps Te then gives the decomposition of the representation ρα.

Further, a complete set R of inequivalent irreducible representations adapted to a

chain of subgroups Gn > Gn−1 > · · · > G0 determines the paths in the Bratteli dia-

gram B of the group algebra chain by drawing M(ρ, γ) arrows from a representation

γ ∈ R of Gi to a representation ρ ∈ R of Gi+1. Then a set of bases for the repre-

sentation spaces of the representations in R determines a system of Gel-fand Tsetlin

bases for the group algebra chain, and so by Theorem A.1 a model representation.


We first prove Theorem 2.7 of Section 2.3.

Theorem A.2 (Theorem 2.7). For B a Bratteli diagram of highest grading at least

n, φ = ι1 t ι2, and ι = ι4 as in Figure 2.16, the bilinear map

ι∗ ◦ φ∗ ◦⊗

: A(Qn0;B)× A(Qn0;B)→ A(Qn0;B)

corresponds to the algebra product on C[Bn] under the isomorphism A(Qn0;B) ∼= C[Bn]

143


of Lemma 2.4.

Proof. For P,Q paths of length n in B starting at the root and ending at the same

vertex, let τPQ ∈ Hom(Qn0;B) such that τPQ(p) = P and τPQ(q) = Q.

For f =∑f |PQ(P,Q) ∈ C[Bn], the isomorphism of Lemma 2.4 maps f to

∑τPQ∈Hom(Qn0;B)

f |PQτPQ.

Now consider multiplication in the path algebra: for f as above and for g ∈ C[Bn],

f ∗ g =∑P,Q′

(∑Q

f |PQg|QQ′)

(P,Q′) −→∑P,Q′

(∑Q

f |PQg|QQ′)τPQ′ .

We show

ι∗φ∗⊗(∑

P,Q

f |PQτPQ,∑P ′,Q′

g|P ′Q′τP ′Q′)

=∑P,Q′

(∑Q


First note that morphisms η ∈ Hom(Qn0 tQn0;B) have form τPQ t τP ′Q′ . Thus,

ι∗φ∗⊗(∑

P,Q

f |PQτPQ,∑P ′,Q′

g|P ′Q′τP ′Q′)

= ι∗φ∗

( ∑P,Q,P ′,Q′

f |PQg|P ′Q′(τPQ t τP ′Q′)

).

Let (fg)|τPQtτP ′Q′ := f |PQg|P ′Q′ and µPQQ′ ∈ Hom(R;B) such that µPQQ′(p) = P,

144


µPQQ′(q) = Q, and µPQQ′(q′) = Q′. Then

ι∗φ∗

(∑(fg)|τPQtτP ′Q′ (τPQ t τP ′Q′)

)= ι∗

∑µPQQ′

(fg)|µPQQ′◦φµPQQ′

= ι∗

∑µPQQ′

(fg)|τPQtτQQ′µPQQ′

=∑µPQQ′

(fg)|τPQtτQQ′ (µPQQ′ ◦ ι)

=∑P,Q′

(∑Q

(fg)|τPQtτQQ′

)τPQ′

=∑P,Q′

(∑Q


Definition A.3. Let B be a locally finite graded quiver, R a graded quiver with finite

subquivers Q1 and Q2, and ιj the natural inclusion Qj ↪→ R, for j = 1, 2. Then define

the general restricted product relative to R,

∗ : A(Q1 ↑ R;B)× A(Q2 ↑ R;B)→ A(Q14Q2 ↑ R;B),

as follows: for f ∈ A(Q1 ↑ R;B), g ∈ A(Q2 ↑ R;B),

∗(f, g) = f ∗ g :=∑

τ∈Hom(Q14Q2↑R;B)

∑η∈Hom(Q1∪Q2↑R;B)

η↓Q14Q2=τ

f |η◦ι1g|η◦ι2τ.

Remark A.2. For ι4 the natural injection Q1 4Q2 ↪→ Q1 ∪Q2, it is easy to check

that this operation is equivalent to (ι4)∗ ◦ (ι1 t ι2)∗ ◦⊗

.

145


Lemma A.2. For B a locally finite graded quiver, R a graded quiver with finite

subquivers Q1 and Q2, and fi ∈ A(Qi;B), the restricted product f1 ∗ f2 requires no

more than

# Hom((Q1 ∪Q2) ↑ R;B)

scalar multiplications and

# Hom((Q1 ∪Q2) ↑ R;B)−# Hom((Q14Q2) ↑ R;B)

scalar additions.

Proof. To compute f1 ∗f2, first compute (f1|η◦ι1)(f2|η◦ι2) for each η ∈ Hom(Q1∪Q2 ↑

R;B). This requires # Hom(Q1 ∪Q2 ↑ R;B) scalar multiplications.

Next note that a scalar addition comes from each pair ηi, ηj ∈ Hom(Q1∪Q2 ↑ R;B)

with

ηi ↓Q14Q2= ηj ↓Q14Q2= τ ∈ Hom(Q14Q2 ↑ R;B);

in total, # Hom((Q1 ∪Q2) ↑ R;B)−# Hom((Q14Q2) ↑ R;B) scalar additions.

Corollary A.1. For w ∈ Wi = C[Bi+ ] ∩ Centralizer(C[Bi− ]), e the identity element

of C[Bn], and quivers G,Gi, F, Fi and maps ξi as in Theorem 2.8,

ξi(w) ∗ ξj(e)

requires no arithmetic operations to compute.

Proof. First note that the value of the identity element at any pair of paths is either

0 or 1. Thus, taking a restricted product with the element of a configuration space

corresponding to the identity will require no multiplications.

146


To find the number of additions, recall that ξi maps Wi to A(Fi;B), for Fi a quiver

with vertices at levels i+ and i−. Since for all 1 ≤ k ≤ n,

e ∈ C[Bk] ∩ Centralizer(C[Bk]),

j+ = j− = n, so Fj consists of a single vertex at level n. Then by Lemma 2.5, the

restricted product ξi(w) ∗ ξj(e) requires no more than

# Hom((Fi ∪ Fj ↑ G;B)−# Hom(Fi4 Fj ↑ G;B) = 0

additions.

Lemma A.3. For B a locally finite graded quiver, R a graded quiver with finite

subquivers Q1 and Q2, and fi ∈ A(Qi;B),

f1 ∗ f2 = f2 ∗ f1

Proof. Clear since Q1 ∪Q2 = Q2 ∪Q1, Q14Q2 = Q24Q1, and the coefficients of f1

and f2 lie in C.

Lemma A.4. Let B be a locally finite graded quiver and R a graded quiver with finite

subquivers Q1, Q2, . . . , Qm such that Qi ∩Qj ∩Qk has no edges for all i 6= j 6= k. Let

Q4i denote the quiver Q14· · ·4Qi and let Q∪i denote the quiver Q1∪ · · · ∪Qi. Then

for fi ∈ A(Qi;B),

f1 ∗ f2 ∗ · · · ∗ fm

is independent of bracketing. Moreover, for τ ∈ Hom(Q4m;B) and ιk the natural

147


injection Qk ↪→ R,

(f1 ∗ f2 ∗ · · · ∗ fm)|τ =∑

η∈Hom(Q∪m↑R;B),η↓Q4m

=τ

m∏k=1

fk|η◦ιk . (A.1)

Proof. We first prove (A.1) inductively, as associativity clearly follows.

For n = 2, (A.1) is the definition of the restricted product f1 ∗ f2.

Now suppose (A.1) holds for n− 1. Since Qi ∩Qj ∩Qk = ∅,

[(Q14 · · · 4Qn−1) ∪Qn] ∩ [Q1 ∪ · · · ∪Qn−1] = Q14 · · · 4Qn−1, (A.2)

and

[(Q14 · · · 4Qn−1) ∪Qn] ∪ [Q1 ∪ · · · ∪Qn−1] = Q1 ∪ · · · ∪Qn. (A.3)

By the induction hypothesis,

(f1 ∗ · · · ∗ fn−1 ∗ fn)|τ =∑

η∈Hom(Q4n−1∪Qn↑R;B)

η↓Q4n

=τ

∑µ∈Hom(Q∪n−1↑R;B)µ↓Q4n−1

=η↓Q4n−1

(n−1∏k=1

fk|µ◦ιk · fn|η◦ιn

).

By (A.2) and (A.3), each choice of µ and η which agree on their intersection, the

subquiver Q4n−1, uniquely determines a morphism γ ∈ Hom(Q∪n ↑ R;B) such that

γ ↓Q14···4Qn= η ↓Q14···4Qn= τ,

γ ◦ ιk = µ ◦ ιk, for 1 ≤ k ≤ n− 1,

γ ◦ ιn = η ◦ ιn.

148

A.3 Quiver Counts

A.3 Quiver Counts

A.3.1 Smoothing Quivers

Definition A.4. A quiver B factors at level i if there are no arrows from a vertex

α ∈ V (B) with gr(α) < i to a vertex β ∈ V (B) with gr(β) > i.

Example A.1. Let B be a Bratteli diagram with highest grading n. Then for all

0 ≤ i ≤ n, B factors at level i.

Definition A.5. Let Q be a quiver with a vertex v that is the head of exactly one

arrow, e1, and the tail of exactly one arrow, e2. To smooth Q at v, remove v and

replace e1 and e2 with an arrow from the tail of e1 to the head of e2. To smooth Q,

smooth Q at all possible v.

Example A.2. The quiver Q′ of Figure A.1 results from smoothing the quiver Q.

Q Q′

012 02

Figure A.1

Lemma A.5. Let B be a graded quiver that factors at level i, R a graded quiver with

subquiver Q, and v a vertex of Q at level i such that Q can be smoothed at v. Let Q′

(respectively R′) be the quiver obtained by smoothing Q (respectively R) at v. Then

# Hom(Q ↑ R;B) = # Hom(Q′ ↑ R′;B).

Proof. Let φ ∈ Hom(Q′ ↑ R′;B) and let f be the arrow in Q′ resulting from smoothing

Q at v. Then f replaced two arrows, e1, e2 in Q, with e+1 = v, e−2 = v. Further,

149

A.3 Quiver Counts

e−1 = f−, e+2 = f+, so φ(f) is a path in B from a vertex α with gr(α) < i to a vertex

β with gr(β) > i. Since B factors at level i, this path contains a vertex, v′, with

gr(v′) = i. Let e′1 be the subpath of f starting at the tail of f and ending at v′.

Similarly, let e′2 be the subpath of f starting at v′ and ending at the head of f .

Denote by φ the morphism in Hom(Q ↑ R;B) such that:

φ(e1) = e′1, φ(e2) = e′2

φ(ei) = φ(ei), for i 6= 1, 2,

φ(vj) = φ(vj), for vj 6= α, β, v′.

Clearly φ→ φ is a bijection.

Corollary A.2. Let B be a Bratteli diagram, R a graded quiver with subquiver Q,

and Q′ (respectively R′) the quiver obtained by smoothing Q (respectively R). Then

# Hom(Q ↑ R;B) = # Hom(Q′ ↑ R′;B).

A.3.2 Properties of Locally Free Quivers

Proposition A.1 (Theore 2.1). Let B be a Bratteli diagram. Then the following

properties are equivalent:

(i) B is locally free.

(ii) For each i and all β ∈ Bi−1, there exists λi ∈ C such that

∑α∈Bi

MB(α, β)dα = λidβ.

150

A.3 Quiver Counts

(iii) For each i, di is an eigenvector of DU .

(iv) For each i there exists λi ∈ C with DU i0 = λiUi−10.

Proof. Statements (ii), (iii), and (iv) are equivalent by definition and Lemma 2.6. For

example:

(iii)⇒ (iv) DU i0 = DUdi−1 = λidi−1 = λiUi−10

(iv)⇒ (iii) DUdi = DU i+10 = λi+1Ui0 = λi+1di

We leave the remaining equivalences of (ii), (iii), and (iv) to the reader.

To show the equivalence of (i) and (iv), recall from Note 2.4 that elements of C[Bi]

correspond to representations of C[Bi], i.e. C[Bi]-modules. Under this identification,

the regular representation of C[Bi] corresponds to the sum

∑α∈Bi

dαα = di ∈ C[Bi].

Since D is restriction ([39][Proposition 2.3.1]), the restriction of the regular represen-

tation of C[Bi] to C[Bi−1] corresponds to Ddi = DUdi−1.

(i)⇒(iv) For B locally free, C[Bi] is free as a module over C[Bi−1] with rank λi ∈ C.

Then the regular representation of C[Bi] decomposes in C[Bi−1] as λi copies of

the regular representation of C[Bi−1]. Thus,

DU i0 = DUdi−1 = Ddi = λidi−1 = λiUi−10.

(iv)⇒(i) DUdi−1 = Ddi = λidi−1 and so dimCC[Bi] = λi dimC C[Bi−1]; hence λi is

151

A.3 Quiver Counts

rational and positive. To show λi integral, let

λi = pq, gcd(p, q) = 1, p, q > 0.

Let m = gcd(dβ) over all β ∈ Bi−1, sodβm

an integer for all β ∈ Bi−1. Then

1

m

p

q

∑β∈Bi−1

dββ =1

mλidi−1

=1

mDUdi−1

=1

m

∑α∈Bi

∑β∈Bi−1

MB(α, β)dαβ

=∑α∈Bi

∑β∈Bi−1

MB(α, β)2dβmβ.

Then the coefficient of β is an integer and thus q|dβm

for all β ∈ Bi. But

m = gcd(dβ), and thus q = 1, making λi an integer.

Theorem A.3 (Theorem 2.13). Let B be a locally free Bratteli diagram and w =

DdnUun · · ·Dd1Uu1 an admissible word in U and D. Then for s =∑n

i=1 ui − di and

α ∈ Bs,

〈w0, α〉 = dα∏i∈S

λbi−ai .

152

A.3 Quiver Counts

Proof. To be admissible, di, ui > 0 for all 1 ≤ i ≤ n and

i∑j=1

dj ≤i∑

j=1

uj.

Let

Sk = {i ∈ S|i ≤k∑j=1

(dj + uj)},

let sk =∑k

i=1 ui − di, and let wk = DdkUuk · · ·Dd1Uu1 . We prove inductively that

wk0 =∏i∈Sk

λbi−ai∑α∈Bsk

dαα.

For k = 1, consider w1 = Dd1Uu1 . Then

S1 = {u1 + 1, u1 + 2 . . . , u1 + d1}

and for all i ∈ S1, bi = u1 and ai = i− u1 − 1. By Proposition 2.1, Lemma 2.6, and

induction,

w10 = Dd1Uu1 0

= Dd1−1λu1Uu1−10

= λu1Dd1−1Uu1−10

...

= λu1λu1−1 · · ·λu1−d1+1Uu1−d1 0

=∏i∈S1

λbi−ai∑α∈Bs1

dαα.

153

A.4 Combinatorial Lemmas for the Weyl Groups and Brauer Algebra

Now suppose true for n− 1. Then

w0 = wn0

= DdnUunwn−10

= DdnUun∏

i∈Sn−1

λbi−aiUsn−1 0

=∏

i∈Sn−1

λbi−aiDdnUun+sn−1 0,

and the same argument as in the base case gives the result.

A.4 Combinatorial Lemmas for the Weyl Groups

and Brauer Algebra

In this section we consider the Bratteli diagrams associated to the Weyl groups Bn

and Dn to provide the bounds used in Theorems 2.2 and 2.3. Note that Lemmas A.6,

A.7, A.9, A.10, A.11, and Corollaries A.3, A.4, and A.5 all hold for n ≥ 2, i ≥ 2.

A.4.1 The Weyl Group Bn

The Bratteli diagram B associated to the chain

C[Bn] > C[Bn−1] > · · · > C

is a generalization of Young’s diagram — inequivalent irreducible representations of

C[Bi] are indexed by pairs of partitions (λ1, λ2), of k and l, respectively, with k+l = i.

Pairs (λ1, λ2), (µ1, µ2) are connected by an edge if either λ1 may be obtained from µ1

154


by adding a box, or if λ2 may be obtained from µ2 by adding a box [77] (see Figure

A.2). Note that this is a multiplicity-free diagram.

· · · ( , )

( , ∅)

(∅, )

( , ∅)

(∅, )

( , ∅)

(∅, )

(∅, ∅)

Figure A.2: C[B2] > C[B1] > C

Lemma A.6. MB(Bi, Bi−2) ≤ 2

Proof. Suppose not. Then since B is multiplicity-free, we must have distinct pairs of

partitions

κ = (κ1, κ2), ρ = (ρ1, ρ2), γ = (γ1, γ2), η = (η1, η2), λ = (λ1, λ2),

as in Figure A.3.

level j

level j − 1

level j − 2

κ

γ ρη

λ

Figure A.3

Pairs of partitions are adjacent in B if one is acquired from the other by adding

a single box; hence either ρ1 = λ1 or ρ2 = λ2. The same holds for η, γ. Similarly,

155


κ1 = ρ1 or κ2 = ρ2 and the same holds for η, γ.

Without loss of generality, we need only consider the following two cases:

Case 1: ρ1, η1, γ1 = λ1.

If κ1 6= λ1, then κ2 = ρ2, γ2, η2, but then ρ = η = γ, a contradiction.

If κ1 = λ1 then κ2 is obtained from λ2 by adding two boxes, which may be done

in at most two ways, so η, γ, ρ are not all distinct, a contradiction.

Case 2: ρ1 = λ1 = η1 and γ2 = λ2.

If κ1 = λ1 then since γ1 6= λ1, we see that κ1 6= γ1. Thus, κ2 = γ2. But then

(κ1, κ2) = (λ1, λ2), a contradiction.

Now if κ1 6= λ1, then κ2 = ρ2, η2, but then ρ = η, a contradiction.

Lemma A.6 is used in the proof of Theorem 2.2 to give a bound on dimA(J ni ↑

G;B) for J ni as in Figure 2.28. The following two lemmas provide a bound for

dimA(Hni ↑ G;B), for Hn

i as in Figure 2.28.

Lemma A.7.

(1) # Hom(Hni ↑ G;B) =

|Bn−1||Bi−1|

# Hom(Hii ↑ G;B)

(2) # Hom(Hii ↑ G;B) =

2(i−1)|Bi−1|+∑

(β1i−1,β

2i−1)=

βi−1∈Bi−1

(jmp(β1i−1)+jmp(β2

i−1))(jmp(β1i−1)+jmp(β2

i−1)+1)d2βi−1

,

where jmp denotes the jump of a partition, i.e, the number of ways to remove a

single box to form a new partition.

156


Proof. To prove (1), first note by Theorem 2.12,

# Hom(Hni ↑ G;B) =

∑αj ,βj∈Bj

(MB(βn−1, βi−1)MB(βi−1, βi−2)MB(αi, αi−1)

MB(αi, βi−1)MB(αi−1, βi−2)dαi−1dβn−1

).

By Corollary 2.2,

∑βn−1∈Bn−1

MB(βn−1, βi−1)dβn−1 = 〈Dn−iUn−10, βi−1〉

= λn−1λn−2 · · ·λidβi−1

=|Bn−1||Bi−1|

dβi−1.

Then

# Hom(Hni ↑ G;B) =

|Bn−1||Bi−1|

∑αj ,βj∈Bj

(MB(βi−1, βi−2)MB(αi, αi−1)

MB(αi, βi−1)MB(αi−1, βi−2)dβi−1dαi−1

)=|Bn−1||Bi−1|

# Hom(Hii ↑ G;B).

To prove (2),

# Hom(Hii ↑ G;B) =

=∑

αj ,βj∈BjMB(βi−1, βi−2)MB(αi, αi−1)MB(αi, βi−1)MB(αi−1, βi−2)dβi−1

dαi−1

=∑

αi−1 6=βi−1

+∑

αi−1=βi−1

,

157


for∑

αi−1 6=βi−1

the sum

∑αj ,βj∈Bjαi−1 6=βi−1

MB(βi−1, βi−2)MB(αi, αi−1)MB(αi, βi−1)MB(αi−1, βi−2)dβi−1dαi−1

and∑

αi−1=βi−1

the sum

∑αj ,βj∈Bjαi−1=βi−1

MB(βi−1, βi−2)2MB(αi, βi−1)2(dβi−1)2.

First suppose αi−1 = (α1i−1, α

2i−1), βi−1 = (β1

i−1, β2i−1) are distinct pairs of partitions.

Then they jointly determine αi = (α1i , α

2i ). Thus,

∑αi−1 6=βi−1

=∑

βi−2∈Bi−2

αi−1 6=βi−1∈Bi−1

MB(βi−1, βi−2)MB(αi−1, βi−2)dαi−1dβi−1

=∑

βi−2∈Bi−2

αi−1,βi−1∈Bi−1

MB(βi−1, βi−2)MB(αi−1, βi−2)dβi−1dαi−1

−

∑βj∈B

αi−1=βi−1

MB(βi−1, βi−2)2(dβi−1)2

158


=|Bi−1||Bi−2|

∑βj ,αj∈Bj

MB(αi−1, βi−2)dβi−2dαi−1

−∑βj∈Bj

αi−1=βi−1

MB(βi−1, βi−2)2(dβi−1)2

=|Bi−1||Bi−2|

∑αi−1∈Bi−1

(dαi−1)2 −

∑βj∈Bj

αi−1=βi−1

MB(βi−1, βi−2)2(dβi−1)2

=|Bi−1|2

|Bi−2|−

∑βi−1∈Bi−1

(jmp(β1i−1) + jmp(β2

i−1))(dβi−1)2

= 2(i− 1)|Bi−1| −∑

βi−1∈Bi−1


i−1))(dβi−1)2. (A.4)

Now suppose αi−1 = βi−1. Then αi is obtained from βi−1 by adding a box to β1i−1

or β2i−1, while βi−2 is obtained from βi−1 by removing a box from β1

i−1 or β2i−1. Thus,

∑αi−1=βi−1

=∑

βi−1∈Bi−1



i−2)+2)(dβi−1)2. (A.5)

Summing equations (A.4) and (A.5), we have

# Hom(Hii ↑ G;B) = 2(i− 1)|Bi−1|

+∑

(β1i−1,β

2i−1)=

βi−1∈Bi−1


i−1))(jmp(β1i−1) + jmp(β2

i−1) + 1)d2βi−1

.

159


Lemma A.8. For any pair of partitions (β1i , β

2i ) with |β1

i |+ |β2i | = i,

(jmp(β1i ) + jmp(β2

i ))(jmp(β1i ) + jmp(β2

i ) + 1) ≤ 6i

Proof. Let k = |β1i |, l = |β2

i |, ak = jmp(β1i ), and al = jmp(β2

i ). Then k + l = i and

by [63, Lemma 5.3],

ak(ak + 1) ≤ 2k, al(al + 1) ≤ 2l.

Then

(jmp(β1i ) + jmp(β2

i ))(jmp(β1i ) + jmp(β2

i ) + 1) = (ak + al)(ak + al + 1)

= ak(ak + 1) + al(al + 1) + 2akal

≤ 2k + 2l + 2(2i)

≤ 6i.

Combining Lemmas A.7 and A.8 gives the following bound:

Corollary A.3. # Hom(Hni ↑ G;B) ≤ 4(i−1)

n|Bn|.

A.4.2 The Weyl Group Dn

The Bratteli diagram B associated to the chain

C[Dn] > C[Dn−1] > · · · > C

is similar to the Bratteli diagram associated to the Weyl group Bn in that irreducible

representations of C[Di] are indexed by pairs of partitions, (λ1, λ2) of k and l, respec-

160


tively, with k + l = i. However, if λ1 6= λ2, the irreducible representation indexed

by (λ1, λ2) is the same as that indexed by (λ2, λ1). If λ1 = λ2 = λ then two distinct

irreducible representations are indexed by the pair (λ, λ), and denoted by (λ, λ)+ and

(λ, λ)− [77] (see Figure A.4). Note that this is a multiplicity-free diagram.

( , )

( , )

( , ∅)

( , ∅)

( , ∅)

( , )+

( , )−

( , ∅)

( , ∅)( , ∅) (∅, ∅)· · ·

Figure A.4: C[D3] > C[D2] > C[D1] > C

Lemma A.9. MB(Di, Di−2) ≤ 3.

Proof. Suppose not. Then since B is multiplicity-free, there exist pairs of partitions

κ = (κ1, κ2), ρ = (ρ1, ρ2), γ = (γ1, γ2), η = (η1, η2), µ = (µ1, µ2), λ = (λ1, λ2),

connected in B as in Figure A.5.

level j

level j − 1

level j − 2

κ

γρη µ

λ

Figure A.5

However, the proof of Lemma A.6 dictates that no three of η, µ, γ, ρ are distinct

161


pairs of partitions. Thus, without loss of generality,

(η1, η2) = (α, α)+, (µ1, µ2) = (α, α)−, (γ1, γ2) = (β, β)+, (ρ1, ρ2) = (β, β)−,

for α, β distinct partitions of j−12

. Then as in the proof of Lemma A.6, either λ1 =

η1 = α or λ2 = η2 = α. Without loss, suppose λ1 = α. Then since α 6= β, λ2 must be

β. However, |λ1|+ |λ2| = |α|+ |β| > j − 2, a contradiction.

Lemma A.9 is used in the proof of Theorem 2.3 to give a bound on dimA(J ni ↑

G;B), for J ni as in Figure 2.28. The following two lemmas provide a bound for

dimA(Hni ↑ G;B), for Hn

i as in Figure 2.28.

Lemma A.10.

(1) # Hom(Hni ↑ G;B) =

|Dn−1||Di−1|

# Hom(Hii ↑ G;B),

(2) for i odd,

# Hom(Hii ↑ G;B) ≤

|Di−1|2

|Di−2|+

∑(α,α)±∈Bi−1

(jmp(α))(jmp(α) + 1)(d(α,α)+)2+

∑(β1i−1,β

2i−1)=

βi−1∈Bi−1


i−1))(jmp(β1i−1) + jmp(β2

i−1) + 1)d2βi−1

,

162


(3) for i even,

# Hom(Hii ↑ G;B) ≤ 2

|Di−1|2

|Di−2|+2

∑(β1i−1,β

2i−1)=

βi−1∈Bi−1


i−1))(2 jmp(β1i−1) + 2 jmp(β2

i−1) + 3)d2βi−1

,

where jmp denotes the jump of a partition, i.e., the number of ways to remove

a single box to form a new partition.

Proof. Part (1) follows from the proof of Lemma A.7.

To prove (2), consider


=∑


dαi−1

=∑

αi−1 6=βi−1

+∑

αi−1 6=βi−1

αi−1=(α,α)±=βi−1

+∑

αi−1=βi−1

,

For

∑αi−1 6=βi−1

:=



,

163


over partitions αi−1 6= βi−1 such that if αi−1 = (α, α)± then βi−1 6= (α, α)±,

∑αi−1 6=βi−1

αi−1:=(α,α)±

=βi−1

:=


αi−1=(α,α)±=βi−1


,

and ∑αi−1:=βi−1

:=∑

αj ,βj∈Bjαi−1=βi−1


As in the proof of Lemma A.7,

∑αi−1 6=βi−1

≤ |Di−1|2

|Di−2|−

∑βi−1∈Bi−1


i−1))(dβi−1)2, (A.6)

the inequality appearing because if βi−1 = (α, α), jmp(α)+jmp(α) is an overestimate

since (α, β) represents the same representation as (β, α) in B. Similarly, the proof of

Lemma A.7 gives

∑αi−1=βi−1

≤∑

βi−1∈Bi−1



i−1)+2)(dβi−1)2. (A.7)

Now suppose αi−1 6= βi−1 and αi−1 = (α, α)± = βi−1. Then

∑αi−1 6=βi−1

αi−1=(α,α)±=βi−1

=∑

αj ,βj∈Bj(α,α)±

MB((α, α)±, βi−2)2MB(αi, (α, α)±)2(d(α,α)±)2

≤∑

(α,α)±∈Bi−1

jmp(α)(jmp(α) + 1)(d(α,α)±)2. (A.8)

164


Summing equations (A.6), (A.7), and (A.8) gives part (2).

To prove (3), note that in this case


∑αi−1 6=βi−1

+∑

αi−1=βi−1

,

since i− 1 is odd so (α, α)± /∈ Bi−1. However, pairs of partitions of this form may be

found at levels i and i− 2.

First suppose αi−1 6= βi−1. Then as in the proof of Lemma A.7 they jointly

determine αi = (α1i , α

2i ). This means that they jointly determine at most two pairs

of partitions (if α1i = α2

i ). Thus

∑αi−1 6=βi−1

≤ 2∑

βi−2∈Bi−2

αi−1 6=βi−1∈Bi−1

MB(βi−1, βi−2)MB(αi−1, βi−2)dαi−1dβi−1

= 2|Di−1|2

|Di−2|− 2

∑βi−1∈Bi−1


i−1))(dβi−1)2, (A.9)

as in the proof of Lemma A.7.

Now suppose αi−1 = βi−1. As before there are jmp(β1i−1) ways to obtain β1

i and

jmp(β2i−1) ways to obtain β2

i , but to account for when β1i = β2

i , we overcount by

multiplying by 2. The same holds for the number of ways to obtain αi−2 from βi−1.

Thus,

∑αi−1βi−1

≤∑

βi−1∈Bi−1

2(jmp(β1i−1) + jmp(β2

i−1))2(jmp(β1i−1) + jmp(β2

i−1) + 2)(dβi−1)2.

(A.10)

Summing equations (A.9) and (A.10) gives part 3.

165


Combining Lemma A.10 with Lemma A.8 gives the following bound:

Corollary A.4. # Hom(Hni ↑ G;B) ≤ 20(i−1)

n|Dn|.

A.4.3 The Brauer Algebra Brn

In the Bratteli diagram B associated to the chain

Brn > Brn−1 > · · · > C(q)

irreducible representations of Bri are indexed by partitions of i − 2k, 0 ≤ k ≤ i/2,

with an edge between ρ ∈ Bi and λ ∈ Bi−1 if ρ is obtained from λ by adding or

removing a box [56] (see Figure A.6). Note that this is a multiplicity-free diagram.

∅

∅· · ·

Figure A.6: Br3 > Br2 > Br1 > C(q)

The following two lemmas provide a bound for dimA(Hni ↑ G;B), for Hn

i as in

Figure 2.43.

Lemma A.11.

(1) # Hom(Hni ↑ G;B) = dim(Brn−1)

dim(Bri−1)# Hom(Hi

i ↑ G;B),

166


(2) # Hom(Hii ↑ G;B)

≤ 2dim(Bri−1)2

dim(Bri−2)+

∑βi−1∈Bi−1

(4 jmp(βi−1)2 + 2 jmp(βi−1) + 1)(dβi−1)2,

where jmp denotes the jump of a partition, i.e, the number of ways to remove a

single box to form a new partition.

Proof. Part (1) follows from the proof of Lemma A.7.

To prove (2), consider

# Hom(Hii ↑ G;B)

=∑


dαi−1

=∑

αi−1 6=βi−1

+∑

αi−1=βi−1

,

for∑

αi−1 6=βi−1

the sum



and∑

αi−1=βi−1

the sum

∑αj ,βj∈Bjαi−1=βi−1


First suppose αi−1 and βi−1 are distinct partitions. Then they jointly determine

αi up to two choices. This is clear if αi−1 and βi−1 both partition k, as they then

167

A.5 Factoring Coset Representatives of GLn(Fq)

they jointly determine exactly one partition of k + 1 and one partition of k − 1.

Now suppose, without loss of generality, that αi−1 is a partition of k while βi−1 is a

partition of k − 2. Then to both be connected to a vertex, αi, at level i, βi−1 must

be obtained from αi−1 by removing two boxes, which can only be done in two ways.

Then as in the proof of Lemma A.7,

∑αi−1 6=βi−1

≤ 2

(dim(Bri−1)2

dim(Bri−2)−∑

jmp(βi−1)(dβi−1)2

). (A.11)

Now suppose αi−1 = βi−1. Then αi is obtained from βi−1 by either adding or

removing a box, and similarly for βi−2. Thus,

∑αi−1=βi−1

=∑

βi−1∈Bi−1

(2 jmp(βi−1) + 1)(2 jmp(βi−1) + 1)(dβi−1)2. (A.12)

Summing equations (A.11) and (A.12) gives part (2).

Combining Lemma A.11 with the proof of Lemma A.8 gives the following bound:

Corollary A.5. # Hom(Hni ↑ G;B) ≤ 16i−17

2n−1dim(Brn).


In this section we provide the set of coset representatives and their factorizations used

in the proof of Theorem 2.4 by developing a correspondence between Gln(q)/Gln−1(q)

and the set

Xn = {(y,x) | x,y ∈ (Fq)n, y ∗ xT = 1}.

168


Define an action of Gln(q) on Xn via A.(y,x) = (yA−1, AxT ). Note that this

action preserves the product y ∗ xT . For 1 = ((0, . . . , 0, 1), (0, . . . , 0, 1)), we show

(Theorem A.4) that

Xn = Orb(1).

Further, note that Gln−1(q), viewed as a subgroup of Gln(q), stabilizes 1, so the

orbit-stabilizer theorem gives a bijection between Xn = Orb(1) and Gln(q)/Gln−1(q)

through the correspondence

g.1←→ gGln−1(q). (A.13)

Thus, writing (x,y) = A1 · · ·Am.1 for each (x,y) ∈ Xn gives a factorization of the

corresponding coset representative. We find a factorization in which each matrix

Ai = A⊕

In−2 for A ∈ Gl2(q).

Lemma A.12. Suppose x,y ∈ (Fq)2 with x1y1 +x2y2 6= 0. Then there exists a matrix

A ∈ Gl2(q) and y′2, x′2 ∈ F×q such that

A.(y,x) = (

(0, y′2

),

(0, x′2

)).

Proof. Since the action of A preserves the product y ∗ xT ,

y′2x′2 = x1y1 + x2y2 6= 0.

Case 1: x1 = 0. Let A =

1 0

y1y2

1

. Note there are q possibilities for A.

169


Case 2: x1 6= 0, y1 = 0. Let A =

1 −x1x2

0 1

. Note there are q − 1 possibilities for A.

Case 3: x1 6= 0, y1 6= 0. Let A =

−x2x11

1 y2y1

. Note there are q2 possibilities for A.

Note further that for z1 := x1y1 and z2 := x2y2 fixed and nonzero,

A =

−x2x11

1 z2z1

x1x2

,

and there are q − 1 possibilities for A.

We use Lemma A.12 to systematically write (y, x) ∈ Xn in form

(y, x) = A1A2 · · ·Ak.((

0, . . . , 0, y′n

),

(0, . . . , 0, x′n

)),

with Ai ∈ Gln(q) and y′nx′n = 1. To do so we first simplify (y, x). Recall that p is the

characteristic of Fq.

Proposition A.2. Let (y, x) ∈ Xn. Then there is a permutation matrix π ∈ Gln(q),

b ∈ F×q , and i ≥ 1 such that π.(y, x) = (y,x) with:

(i) z1 + · · ·+ zj 6= 0 for all i ≤ j ≤ n, where zi := xiyi,

(ii) z1 = · · · = zi = b,

(iii) p|(i− 1).

170


Proof. Let z = (z1, . . . , zn) := (x1y1 . . . , xnyn). Note that y ∗ xT = z1 + · · · + zn =

1 6= 0. Let j be an index (if it exists) such that y ∗ xT − zj 6= 0. Note that for a

permutation matrix π, (πT )−1 = π, so the action of π on y, x) permutes x and y in

the same way and we may equivalently consider the action of π on z. Permute z to

make zj the last entry, then delete zj to produce a vector of length n−1. Repeat until

no such index exists, and let i be the length of the resultant vector, z. Then clearly

z1 + · · · + zj 6= 0 for all i ≤ j ≤ n. Further, z1 + · · · + zi − zk = 0 for all 1 ≤ k ≤ i;

in particular, z1 = · · · = zi = b ∈ F×q . Finally note that z1 + · · ·+ zi−1 = (i− 1)b = 0

and so p|(i− 1).

In light of Proposition A.2, let

Si(n) = {(y,x) ∈ Xn| (y,x) satisfies (i) and (ii) of Proposition A.2}.

Note that if p|i, Si(n) = ∅.

Theorem A.4. For p 6= 2 and (y,x) ∈ Si(n), there exist invertible matrices

uj, u′j, vj, tj ∈ Glj(q) ∩ Centralizer(Glj−2(q)),

and a scalar matrix ε such that

(y,x) = (u2 · · ·up−1u′p+1tpup+1 · · ·u2p−1u

′2p+1t2pu2p+1 · · ·uivi+1 · · · vnε).1.

Proof. Let (y,x) ∈ Si(n) and let i = mp + l, for m ≥ 1. For z = (z1, . . . , zn) :=

(x1y1, . . . xnyn),

z =

(b, . . . , b, zi+1, . . . , zn

).

171


Note that z1 + z2 = 2b 6= 0 and so by Lemma A.12, there is a matrix A ∈ Gl2(q) such

that A.(

(y1, y2

),

(x1, x2

)) = (

(0, y′2

),

(0, x′2

)) with y′2x

′2 = 2b. Let

u−12 =

A

1

. . .

1

∈ Gln(q).

Then u−12 .(y,x) = (

(0, y′2, y3, . . . , yn

),

(0, x′2, x3, . . . , xn

)) with new z-vector

(0, 2b, b, . . . , b, zi+1, . . . , zn

).

Provided zj + zj+1 6= 0, we repeat this process, defining matrices u−13 , u−1

4 , . . . , u−1j .

This affects the z-vector as follows:(b, b, b, b, . . . , b, zi+1, . . . , zn

)↓ u−1

2(0, 2b, b, b, . . . , b, zi+1, . . . , zn

)↓ u−1

3(0, 0, 3b, b, . . . , b, zi+1, . . . , zn

)↓ u−1

4

...

↓ up−1−1(0, . . . , 0, (p− 1)b, b, b, b, . . . , b, zi+1, . . . , zn

).

172


Since zp−1 + zp = pb = 0, instead, define (u′p+1)−1 as above and consider the permu-

tation matrix t−1p corresponding to the transposition (p− 1 p):

↓ (u′p+1)(0, . . . , 0, (p− 1)b, 0, 2b, b, . . . , b, zi+1, . . . , zn

)↓ t−1

p(0, . . . , 0, (p− 1)b, 2b, b, b, . . . , b, zi+1, . . . , zn

)↓ u−1

p+1(0, . . . , 0, 0, (p+ 1)b = b, b, b, . . . , b, zi+1, . . . , zn

)↓ u−1

p+2

...

Repeat this process until the matrix u−1mp+l = u−1

i is defined:

...

↓ u−1mp+l(

0, . . . , 0, z1 + · · ·+ zi, zi+1, . . . , zn

).

173


Since z1 + · · ·+ zj 6= 0 for all i ≤ j ≤ n, use Lemma A.12 to define:

v−1j =

1

. . .

1

A

1

. . .

1

,

with A in the (j − 1, j)x(j − 1, j)th subblock to continue the process:

↓ v−1i+1(

0, . . . , 0, z1 + · · ·+ zi+1, zi+2, . . . , zn

)↓ v−1

i+2

...

↓ v−1n(

0, . . . , 0, z1 + · · ·+ zn

).

Note that z1 + · · ·+ zn = 1. Thus,

v−1n · · · v−1

i+1u−1i · · ·u−1

2p+1t−12p (u′2p+1)−1u−1

2p−1 · · ·u−1p+1t

−1p (u′p+1)−1u−1

p−1 · · ·u−12 .(y,x)

= (

(0, . . . , 0, y′n

),

(0, . . . , 0, x′n

)),

with y′nx′n = 1.

174


For i < p analogous arguments apply without needing the matrices tp.

Multiplying by a scalar matrix, ε−1, and noting that each matrix is invertible and

that vj, uj, u′j, tj ∈ Glj(q) ∩ Centralizer(Glj−2(q)) gives the result.

Remark A.3. By Lemma A.12, there are (q − 1) possibilities for each uj and q2

possibilities for each vj. Since ε is a scalar matrix, there are (q− 1) possibilities for ε.

By Proposition A.2,

Xn =⋃π∈Sn

⋃1≤i≤np|(i−1)

πSi(n)

and so by Correspondence A.13 a complete set of coset representatives for

Gln(q)/Gln−1(q) is contained in {πsi| 1 ≤ i ≤ n, p | (i− 1), si ∈ Si(n)}, with each si

of form:

si = u2 · · ·up−1u′p+1tpup+1 · · ·uivi+1 · · · vn.

Finally, consider the p = 2 case.

Theorem A.5. For p = 2, i ≥ 3 odd, (y,x) ∈ Si(n), there exist invertible matrices

aj, bj, cj, vj ∈ Glj(q) ∩ Centralizer(Glj−2(q))

such that

(y,x) = a3b2c3 · · · aibi−1civi+1 · · · vn.1.

Proof. Let (y,x) ∈ Si(n). For z = (z1, . . . , zn) := (x1y1, . . . xnyn), Proposition A.2

gives

z =

(1, . . . , 1, zi+1, . . . , zn

).

First note that since p - i, i is odd, and for i = 1, Lemma A.12 applies as in the

175


proof of Theorem A.4, so assume i ≥ 3. Also note that for j ≤ i, zj = 1 and so

neither xj nor yj are zero. Then for

Aj =

1yjyj−1

0 1

, Bj =

0 1

1yjyj−1

, Cj =

yj−2

yj1

1 0

,

let a−1j (respectively b−1

j , c−1j )∈ Gln(q) have 1’s along the diagonal and Aj (respectively

Bj, Cj) in the (j − 1, j)x(j − 1, j)th subblock. Then for j ≤ i,

c−1j b−1

j−1a−1j .(x,y) =

(

(y1, . . . , yj−3, 0, 0, yj−2, yj+1, . . . , yn

),

(x1, . . . , xj−3, 0, 0, xj−2, xj+1, . . . , xn

)).

Further, note that a−1j , b−1

j , c−1j are invertible for each 1 ≤ j ≤ i. Then for v−1

j as

in the proof of Theorem A.4,

v−1n · · · v−1

i+1c−1i b−1

i−1a−1i c−1

i−2b−1i−3a

−1i−2 · · · c−1

3 b−12 a−1

3 .(y,x) = 1

which proves the theorem.

Note that there are (q− 1) choices for aj and bj, that cj is completely determined

by aj and bj, and that there are q2 choices for vj.

176

Appendix B

Appendix

B.1 Example of Walk in BMW3

Example B.1. In BMW3,

B3 = R ∪ E1 ∪ E2 ∪ E3,

for

R = {Tid, Tr1 , Tr2 , Tr1Tr2 , Tr2Tr1 , Tr1Tr2Tr1}, E1 = {Te1 , Tr2Te1 , Te2Te1},

E2 = {Te2 , Tr1Te2 , Te1Te2}, E3 = {Te1Tr2 , Tr2Te1Tr2 , Te2Te1Tr2}.

The Markov chain K1 has form

R⊕

E1

⊕E2

⊕E3,

for

177

B.2 Symmetric Group Elements

R =

id r1 r2 r1r2 r2r1 r1r2r1

id 0 θ 0 0 0 0

r1 1 1− θ 0 0 0 0

r2 0 0 0 θ 0 0

r1r2 0 0 1 1− θ 0 0

r2r1 0 0 0 0 0 θ

r1r2r1 0 0 0 0 1 1− θ

,

E1 =

e1 r2e1 e2e1

e1 1 0 0

r2e1 0 0 θ

e2e1 0 1 1− θ

, E2 =

e2 r1e2 e1e2

e2 0 θ 0

r1e2 1 1− θ 0

e1e2 0 0 1

,

E3 =

e1r2 r2e1r2 e2e1r2

e1r2 1 0 0

r2e1r2 0 0 θ

e2e1r2 0 1 1− θ

.


Lemma B.1. Let X1 be a communication class of K whose elements have at least

two lower horizontal edges. Then there exist enough sx ∈ Sn with s2x 6= id to associate

a distinct sx to each x ∈ X1 such that sx 6= s−1y for any y ∈ X1.

Proof. The size of a communication class is determined by the number of lower hor-

izontal edges of its elements. Let Xi be the communication class of an element xi

178


with m lower horizontal edges. Then a simple counting argument gives:

|Xi| = (n− 2m)!m−1∏j=0

(n− 2j

k

).

In particular, for xi, xj ∈ Bn,

|Xi| > |Xj| ⇐⇒ xi has fewer lower horizontal edges than xj.

Note that if xi has exactly one lower horizontal edge,

|Xi| =n!

2=|Sn|

2,

and so Sn cannot contain enough elements of order greater than 2 to make the as-

sociations required by the lemma, as we need 2|Xi| elements of order greater than

2.

Now let xi have exactly two lower horizontal edges. Then for all xj ∈ Bn with at

least two lower horizontal edges,

|Xj| ≤ |Xj| = (n− 4)!

(n

2

)(n− 2

2

)=n!

8=|Sn|

8,

and so

2|Xj| ≤|Sn|

2. (B.1)

Let Tn be the set of elements of Sn of order 2. Then by Equation B.1 we need

show

|Sn|2≤ |Sn| − |Tn|,

179


in other words, that |Tn| ≤ |Sn|2.

But

|Tn| =

n2∑

k=1

n!

(n− 2k)!k!2kif n even ,

n−12∑

k=1

n!

(n− 2k)!k!2kif n odd ,

and so |Tn| < |Sn|2

for n > 4. As the only communication classes when n < 4

correspond to elements with fewer than 2 lower horizontal edges, this proves the

lemma.

Finally, for xi with exactly one lower horizontal edge, while Xi contains too many

elements to make the associations of Lemma B.1, note that each y ∈ Xi can be viewed

as an element, y′ of Bn+1 by adding a verticle edge to the end of the diagram. Then

to analyze [K]i, let X′i = {y′ | y ∈ Xi} and let K ′ be the matrix of K with respect to

Bn+1. Then since [K ′]i = [K]i, we can analyze this case by considering K ′.

180

Bibliography

[1] S. Ariki, N. Jacon, and C. Lecouvey, The modular branching rule for affine Hecke

algebras of type A, Adv. Math. 228 (2011), no. 1, 481–526.

[2] L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform

pure or applied mathematics?, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6,

847–897.

[3] D. Barros, S. Wilson, and J. Kahn, Comparison of orthogonal frequency-division

multiplexing and pulse-amplitude modulation in indoor optical wireless links,

IEEE Trans. Commun. 60 (2012), no. 1, 153–163.

[4] L. Beckett and P. Diaconis, Spectral analysis for discrete longitudinal data, Adv.

Math. 103 (1994), no. 1, 107–128.

[5] G. Benkart, A. Ram, and C.L. Shader, Tensor product representations for or-

thosymplectic Lie superalgebras, J. Pure Appl. Algebra 130 (1998), no. 1, 1–48.

[6] T. Beth, On the computational complexity of the general discrete Fourier trans-

form, Theoret. Comput. Sci. 51 (1987), no. 3, 331–339.

[7] J. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans.

Amer. Math. Soc. 313 (1989), no. 1, 249–273.

181

BIBLIOGRAPHY

[8] O. Bratteli, Inductive limits of finite dimensional C∗-algebras, Trans. Amer.

Math. Soc. . 171 (1972), 195–234.

[9] P. Burgisser, M. Clausen, and M. Shokrollahi, Algebraic Complexity Theory,

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of

Mathematical Sciences], vol. 315, Springer-Verlag, Berlin, 1997, With the col-

laboration of Thomas Lickteig.

[10] Y. Cai, How rates of convergence for Gibbs fields depend on the interaction

and the kind of scanning used, Markov processes and controlled Markov chains

(Changsha, 1999), Kluwer Acad. Publ., Dordrecht, 2002, pp. 489–498.

[11] K. Cannon, R. Cariou, A. Chapman, M. Crispin-Ortuzar, N. Fotopoulos, M. Frei,

C. Hanna, E. Kara, D. Keppel, L. Liao, S. Privitera, A. Searle, L. Singer, and

A. Weinstein, Toward early-warning detection of gravitational waves from com-

pact binary coalescence, The Astrophysical Journal 748 (2012), no. 2, 136.

[12] D. Cartwright and W. Woess, Isotropic random walks in a building of type Ad,

Math. Z. 247 (2004), no. 1, 101–135.

[13] M. Clausen, Fast generalized Fourier transforms, Theoret. Comput. Sci. 67

(1989), no. 1, 55–63.

[14] M. Clausen and U. Baum, Fast Fourier Transforms, Bibliographisches Institut,

Mannheim, 1993.

[15] M. Clausen and R. Kakarala, Computing Fourier transforms and convolutions of

Sn−1-invariant signals on Sn in time linear in n, Appl. Math. Lett. 23 (2010),

no. 2, 183–187.

182

BIBLIOGRAPHY

[16] A. Cohen, D. Gijsbers, and D. Wales, BMW algebras of simply laced type, J.

Algebra 286 (2005), no. 1, 107–153.

[17] J. Cooley and J. Tukey, An algorithm for the machine calculation of complex

Fourier series, Math. Comp. 19 (1965), 297–301.

[18] C. Curtis and I. Reiner, Methods of Representation Theory: with Applications to

Finite Groups and Orders. Vols. I and II, Wiley Classics Library, John Wiley &

Sons, Inc., New York, 1990, Reprint of the 1981 original.

[19] A. Danelakis, M. Mitrouli, and D. Triantafyllou, Blind image deconvolution using

a banded matrix method, Numer. Algorithms 64 (2013), no. 1, 43–72.

[20] Z. Daugherty and R. Orellana, The quasi-partition algebra, J. Algebra 404

(2014), 124–151.

[21] Z. Daugherty, A. Ram, and R. Virk, Affine and degenerate affine BMW algebras:

actions on tensor space, Selecta Math. (N.S.) 19 (2013), no. 2, 611–653.

[22] , Affine and degenerate affine BMW algebras: the center, Osaka J. Math.

51 (2014), no. 1, 257–283.

[23] P. Diaconis, Average running time of the fast Fourier transform, J. Algorithms

1 (1980), 187–208.

[24] , Group Representations in Probability and Statistics, Institute of Mathe-

matical Statistics Lecture Notes—Monograph Series, 11, Institute of Mathemat-

ical Statistics, Hayward, CA, 1988.

183

BIBLIOGRAPHY

[25] P. Diaconis and A. Ram, Analysis of systematic scan Metropolis algorithms using

Iwahori-Hecke algebra techniques, Michigan Math. J. 48 (2000), 157–190.

[26] P. Diaconis and D. Rockmore, Efficient computation of the Fourier transform

on finite groups, J. Amer. Math. Soc. 3 (1990), no. 2, 297–332.

[27] P. Diaconis and L. Saloff-Coste, What do we know about the Metropolis algo-

rithm?, J. Comput. System Sci. 57 (1998), no. 1, 20–36, 27th Annual ACM

Symposium on the Theory of Computing (STOC’95) (Las Vegas, NV).

[28] Y. Diao, C. Ernst, and U. Ziegler, Generating large random knot projections,

Physical and Numerical Models in Knot Theory, Ser. Knots Everything, vol. 36,

World Sci. Publ., Singapore, 2005, pp. 473–494.

[29] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general

linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57–82.

[30] M. Dyer, L. Goldberg, and M. Jerrum, Systematic scan for sampling colorings,

Ann. Appl. Probab. 16 (2006), no. 1, 185–230.

[31] D. Elliott and K. Rao, Fast Transforms: Algorithms, Analyses, Applications,

Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

[32] G. Elliott, On the classification of inductive limits of sequences of semisimple

finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44.

[33] W. Feller, An Introduction to Probability Theory and its Applications. Vol. I,

Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968.

184

BIBLIOGRAPHY

[34] G. Fishman, Coordinate selection rules for Gibbs sampling, Ann. Appl. Probab.

6 (1996), no. 2, 444–465.

[35] D. Flath, T. Halverson, and K. Herbig, The planar rook algebra and Pascal’s

triangle, Enseign. Math. (2) 55 (2009), no. 1-2, 77–92.

[36] P. Gabriel, Unzerlegbare darstellungen I, Manuscripta Math. 6 (1972), no. 1,

71–103 (English).

[37] I. Gel′fand and M. Cetlin, Finite-dimensional representations of the group of

unimodular matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 825–828.

[38] D. Gijsbers, BMW Algebras of Simply Laced Type, Universiteitsdrukkerij Tech-

nische Universiteit Eindhoven, 2005, Thesis (Ph.D.)–Technische Universiteit

Eindhoven.

[39] F. Goodman, P. de la Harpe, and V. Jones, Coxeter Graphs and Towers of Alge-

bras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-

Verlag, New York, 1989.

[40] F. Goodman and H. Hauschild, Affine Birman-Wenzl-Murakami algebras and

tangles in the solid torus, Fund. Math. 190 (2006), 77–137.

[41] F. Goodman and H. Wenzl, Iwahori-Hecke algebras of type A at roots of unity,

J. Algebra 215 (1999), no. 2, 694–734.

[42] R. Goodman and N. Wallach, Representations and Invariants of the Classical

Groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge

University Press, Cambridge, 1998.

185

BIBLIOGRAPHY

[43] C. Grood, The rook partition algebra, J. Combin. Theory Ser. A 113 (2006),

no. 2, 325–351.

[44] T. Halverson and E. delMas, Representations of the Rook-Brauer algebra, Comm.

Algebra 42 (2014), no. 1, 423–443.

[45] T. Halverson and A. Ram, Characters of algebras containing a Jones basic

construction: the Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras,

Adv. Math. 116 (1995), no. 2, 263–321.

[46] T. Halverson and M. Reeks, Gelfand models for diagram algebras, J. Algebraic

Combin. 41 (2015), no. 2, 229–255.

[47] M. Heideman, D. Johnson, and C. Burrus, Gauss and the history of the fast

Fourier transform, Arch. Hist. Exact Sci. 34 (1985), no. 6, 265–277.

[48] M. Jalsenius and K. Pedersen, A systematic scan for 7-colourings of the grid,

Internat. J. Found. Comput. Sci. 19 (2008), no. 6, 1461–1477.

[49] G. James and A. Kerber, The Representation Theory of the Symmetric Group,

Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Pub-

lishing Co., Reading, Mass., 1981.

[50] M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter

equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252.

[51] S. Johnson and M. Frigo, A modified split-radix FFT with fewer arithmetic op-

erations, IEEE Trans. Signal Process. 55 (2007), no. 1, 111–119.

186

BIBLIOGRAPHY

[52] H. Kfir and I. Kanter, Parallel versus sequential updating for belief propagation

decoding, Ann. Appl. Probab. 16 (2006), no. 1, 185–230.

[53] I. Kondor, Group Theoretical Methods in Machine Learning, 2008, Thesis

(Ph.D.)–Columbia University.

[54] P. Kostelec and D. Rockmore, FFTs on the rotation group, J. Fourier Anal. Appl.

14 (2008), no. 2, 145–179.

[55] G. Lebanon and J. Lafferty, Cranking: Combining rankings using conditional

probability models on permutations, Proceedings of the 19th International Con-

ference on Machine Learning, 2002, pp. 363–370.

[56] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible represen-

tations of centralizer algebras: The Brauer,Birman-Wenzl, and type A Iwahori-

Hecke algebras, Adv. Math. 125 (1997), 1–94.

[57] D. Levin, Y. Peres, and E. Wilmer, Markov Chains and Mixing Times, American

Mathematical Society, Providence, RI, 2009.

[58] J. Liu, Monte Carlo Strategies in Scientific Computing, Springer Series in Statis-

tics, Springer, New York, 2008.

[59] T. Lundy and J. Van Buskirk, A new matrix approach to real FFTs and convo-

lutions of length 2k, Computing 80 (2007), no. 1, 23–45.

[60] J. Ma, Components of random links, J. Knot Theory Ramifications 22 (2013),

no. 8, 1350043, 11.

187

BIBLIOGRAPHY

[61] M. Malandro, Fast Fourier transforms for finite inverse semigroups, J. Algebra

324 (2010), no. 2, 282–312.

[62] D. Maslen, Efficient computation of Fourier transforms on compact groups, J.

Fourier Anal. Appl. 4 (1998), no. 1, 19–52.

[63] , The efficient computation of Fourier transforms on the symmetric group,

Math. Comp. 67 (1998), no. 223, 1121–1147.

[64] D. Maslen and D. Rockmore, Adapted diameters and FFTs on groups, Proc. 6th

ACM-SIAM SODA, ACM, 1995, pp. 253–262.

[65] , Separation of variables and the computation of Fourier transforms on

finite groups. I, J. Amer. Math. Soc. 10 (1997), no. 1, 169–214.

[66] , Double coset decompositions and computational harmonic analysis on

groups, Journal of Fourier Analysis and Applications 6 (2000), no. 4, 349–388.

[67] , The Cooley-Tukey FFT and group theory, Notices of the Amer. Math.

Soc. 48 (2001), no. 10, 1151–1160.

[68] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, Equation

of state calculations by fast computing machines, J. Chem. Phys. 21 (1953),

no. 1087, 1087–1092.

[69] H. Morton and A. Wasserman, A basis for the Birman-Wenzl algebra, (1989,

revised 2000), 29 pp., unpublished manuscript, arXiv:1012.3116.

[70] J. Murakami, The Kauffman polynomial of links and representation theory, Osaka

J. Math. 24 (1987), no. 4, 745–758.

188

BIBLIOGRAPHY

[71] A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing (2nd

ed.), Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1999.

[72] J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier

(Grenoble) 57 (2007), no. 2, 379–419.

[73] K. Pedersen, Dobrushin conditions for systematic scan with block dynamics,

Mathematical foundations of computer science 2007, Lecture Notes in Comput.

Sci., vol. 4708, Springer, Berlin, 2007, pp. 264–275.

[74] , On Systematic Scan, 2008, Thesis (Ph.D.)–The University of Liverpool.

[75] , On systematic scan for sampling H-colorings of the path, Random

Struct. Algor. 35 (2009), no. 1, 15–41.

[76] A. Ram, Representation Theory and Character Theory of Centralizer Algebras,

ProQuest LLC, Ann Arbor, MI, 1991, Thesis (Ph.D.)–University of California,

San Diego.

[77] , Seminormal representations of Weyl groups and Iwahori-Hecke algebras,

Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133.

[78] A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and

Clifford theory, A tribute to C. S. Seshadri (Chennai, 2002), Trends Math.,

Birkhauser, Basel, 2003, pp. 428–466.

[79] M. Richey, The evolution of Markov chain Monte Carlo methods, Amer. Math.

Monthly 117 (2010), no. 5, 383–413.

189

BIBLIOGRAPHY

[80] D. Rockmore, Some applications of generalized FFTs, Groups and computation,

II (New Brunswick, NJ, 1995), DIMACS Ser. Discrete Math. Theoret. Comput.

Sci., vol. 28, Amer. Math. Soc., Providence, RI, 1997, pp. 329–369.

[81] , The FFT: An algorithm the whole family can use, Computing in Science

and Eng. 2 (2000), no. 1, 60–64.

[82] M. Rørdam and E. Størmer, Classification of Nuclear C∗-algebras. Entropy in

Operator Algebras, Encyclopaedia of Mathematical Sciences, vol. 126, Springer-

Verlag, Berlin, 2002.

[83] H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser.

A 111 (2005), no. 1, 78–88.

[84] S. Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 42

(1978), no. 4, 279–292.

[85] I. Schur, Uber die rationalen darstellungen der allgemeinen linearen gruppe,

Reprinted in I. Schur, Gesammelte Abhandlungen, Vol. III, pp. 68-85, Springer-

Verlag, Berlin, 1973.

[86] J. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York,

1977, Translated from the second French edition by Leonard L. Scott, Graduate

Texts in Mathematics, Vol. 42.

[87] R. Stanley, Differential posets, J. Amer. Math. Soc. 1 (1988), no. 4, 919–961.

[88] , Variations on differential posets, Invariant theory and tableaux (Min-

neapolis, MN, 1988), IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990,

pp. 145–165.

190

BIBLIOGRAPHY

[89] R. Tolimieri, M. An, and C. Lu, Algorithms for Discrete Fourier Transform and

Convolution, second ed., Signal Processing and Digital Filtering, Springer-Verlag,

New York, 1997.

[90] E. Trachtenberg and M. Karpovsky, Filtering in a communication channel by

Fourier transforms over finite groups, Spectral techniques and fault detection,

Notes Rep. Comput. Sci. Appl. Math., vol. 11, Academic Press, Orlando, FL,

1985, pp. 179–216.

[91] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Fron-

tiers in Applied Mathematics, vol. 10, Society for Industrial and Applied Math-

ematics (SIAM), Philadelphia, PA, 1992.

[92] H. Wenzl, Hecke algebras of type An and subfactors, Invent. Math. 92 (1988),

no. 2, 349–383.

[93] , On the structure of Brauer’s centralizer algebras, Ann. of Math. (2) 128

(1988), no. 1, 173–193.

[94] A. Willsky, On the algebraic structure of certain partially observable finite-state

Markov processes, Inform. and Control 38 (1978), no. 2, 179–212.

[95] A. Young, On quantitative substitutional analysis, Proc. London Math. Soc. 31

(1929), no. 2, 273–288.

[96] P. Zinn-Justin, Conjectures on the enumeration of alternating links, Physical

and numerical models in knot theory, Ser. Knots Everything, vol. 36, World Sci.

Publ., Singapore, 2005, pp. 597–606.

191

BIBLIOGRAPHY

[97] P. Zizler, Some remarks on the non-abelian Fourier transform in crossover de-

signs in clinical trials, Appl. Math. 5 (2014), 917–927.

192

generalized fourier transforms and their applications a thesis submitted to the

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