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Tridiagonal pairs in algebraic graph theory

Paul Terwilliger

University of Wisconsin-Madison

Paul Terwilliger Tridiagonal pairs in algebraic graph theory

Contents

Part I: The subconstituent algebra of a graph

• The adjacency algebra

• The dual adjacency algebra

• The subconstituent algebra

• The notion of a dual adjacency matrix

Part II: Tridiagonal pairs of linear transformations

• the eigenvalues and dual eigenvalues

• the shape

• the tridiagonal relations

• the parameter array

• A classification of TD pairs over an algebraically closed field

Part I. The subconstituent algebra of a graph

Let X denote a nonempty finite set.

MatX (C) denotes the C-algebra consisting of the matrices over Cthat have rows and columns indexed by X .

V = CX denotes the vector space over C consisting of columnvectors with rows indexed by X .

MatX (C) acts on V by left multiplication.

Preliminaries, cont.

Endow V with a Hermitean inner product

〈u, v〉 = utv (u, v ∈ V )

For each x ∈ X let x denote the vector in V that has a 1 incoordinate x and 0 in all other coordinates.

Observe {x |x ∈ X} is an orthonormal basis for V .

The graph Γ

Let Γ = (X ,R) denote a finite, undirected, connected graph,without loops or multiple edges, with vertex set X , edge set R, andpath-length distance function ∂.

For an integer i ≥ 0 and x ∈ X let

Γi (x) = {y ∈ X |∂(x , y) = i}

We abbreviate Γ(x) = Γ1(x).

The graphs of interest

Our main case of interest is when Γ is “highly regular” in a certainway.

A good example to keep in mind is the D-dimensionalhypercube, also called the binary Hamming graph H(D, 2).

Note that H(2, 2) is a 4-cycle; this will be used as a runningexample.

The adjacency matrix

Let A ∈ MatX (C) denote the (0, 1)-adjacency matrix of Γ.

For x ∈ X ,

Ax =∑

y∈Γ(x)

Example

For H(2, 2),

0 1 1 01 0 0 11 0 0 10 1 1 0

The adjacency algebra M

Let M denote the subalgebra of MatX (C) generated by A.

M is called the adjacency algebra of Γ.

M is commutative and semisimple.

Example

For H(2, 2), M has a basis

I ,A, J

where the matrix J has every entry 1.

The primitive idempotents of Γ

Since M is semisimple it has basis {Ei}di=0 such that

EiEj = δijEi (0 ≤ i , j ≤ d),

I =d∑

We call {Ei}di=0 the primitive idempotents of Γ.Write

A =d∑

θiEi .

For 0 ≤ i ≤ d the scalar θi is the eigenvalue of A for Ei .

The primitive idempotents, cont.

Example

For H(2, 2) we have θ0 = 2, θ1 = 0, θ2 = −2. Moreover

E0 = 1/4J,

E1 = 1/4

2 0 0 −20 2 −2 00 −2 2 0−2 0 0 2

E2 = 1/4

1 −1 −1 1−1 1 1 −1−1 1 1 −11 −1 −1 1

The eigenspaces of A

The vector space V decomposes as

V =d∑

EiV (orthogonal direct sum)

For 0 ≤ i ≤ d the space EiV is the eigenspace of A associatedwith the eigenvalue θi .

the matrix Ei represents the orthogonal projection onto EiV .

The dual primitive idempotents of Γ

Until further noticefix x ∈ X .

We call x the base vertex.

Define D = D(x) by

D = max{∂(x , y) | y ∈ X}

We call D the diameter of Γ with respect to x .

For 0 ≤ i ≤ D let E ∗i = E ∗i (x) denote the diagonal matrix inMatX (C) with (y , y)-entry

(E ∗i )yy =

{1, if ∂(x , y) = i ;

0, if ∂(x , y) 6= i .(y ∈ X ).

The dual primitive idempotents, cont.

Example

For H(2, 2),

E ∗0 =

1 0 0 00 0 0 00 0 0 00 0 0 0

, E ∗1 =

0 0 0 00 1 0 00 0 1 00 0 0 0

E ∗2 =

0 0 0 00 0 0 00 0 0 00 0 0 1

The dual primitive idempotents, cont.

The {E ∗i }Di=0 satisfy

E ∗i E ∗j = δijE∗i (0 ≤ i , j ≤ D),

I =D∑i=0

E ∗i .

We call {E ∗i }Di=0 the dual primitive idempotents of Γ withrespect to x .

The dual adjacency algebra

The {E ∗i }Di=0 form a basis for a semisimple commutativesubalgebra of MatX (C) denoted M∗ = M∗(x).

We call M∗ the dual adjacency algebra of Γ with respect to x .

The subconstituents of Γ

The vector space V decomposes as

V =D∑i=0

E ∗i V (orthogonal direct sum)

The above summands are the common eigenspaces for M∗.

These eigenspaces have the following combinatorial interpretation.For 0 ≤ i ≤ D,

E ∗i V = Span{y |y ∈ Γi (x)}

We call E ∗i V the ith subconstituent of Γ with respect to x .

The matrix E ∗i represents the orthogonal projection onto E ∗i V .

The subconstituent algebra T

So far we defined the adjacency algebra M and the dual adjacencyalgebra M∗. We now combine M and M∗ to get a larger algebra.

Definition

(Ter 92) Let T = T (x) denote the subalgebra of MatX (C)generated by M and M∗. T is called the subconstituent algebraof Γ with respect to x .

T is finite-dimensional.

T is noncommutative in general.

T is semisimple

T is semi-simple because it is closed under the conjugate-transposemap.

So by the Wedderburn theory the algebra T is isomorphic to adirect sum of matrix algebras.

Example

For H(2, 2),

T ' Mat3(C)⊕ C.

Moreover dim(T ) = 10.

T is semisimple, cont.

Example

(Junie Go 2002) For H(D, 2),

T ' MatD+1(C)⊕MatD−1(C)⊕MatD−3(C)⊕ · · ·

Moreover dim(T ) = (D + 1)(D + 2)(D + 3)/6.

T -modules

We mentioned that T is closed under the conjugate-transposemap.

So for each T -module W ⊆ V , its orthogonal complement W⊥ isalso a T -module.

Therefore V decomposes into an orthogonal direct sum ofirreducible T -modules.

Problem

(i) How does the above decomposition reflect the combinatorialproperties of Γ? (ii) For which graphs Γ is the abovedecomposition particulary nice?

The dual adjacency matrix

We now describe a family of graphs for which the irreducibleT -modules are nice.

These graphs possess a certain matrix called a dual adjacencymatrix.

To motivate this concept we consider some relations in T .

Some relations in T

By the triangle inequality

AE ∗i V ⊆ E ∗i−1V + E ∗i V + E ∗i+1V (0 ≤ i ≤ D),

where E ∗−1 = 0 and E ∗D+1 = 0.

This is reformulated as follows.

For 0 ≤ i , j ≤ D,

E ∗i AE ∗j = 0 if |i − j ] > 1.

Definition

Referring to the graph Γ, consider a matrix A∗ ∈ MatX (C) thatsatisfies both conditions below:

(i) A∗ generates M∗;

(ii) For 0 ≤ i , j ≤ d ,

EiA∗Ej = 0 if |i − j ] > 1.

We call A∗ a dual adjacency matrix (with respect to x and thegiven ordering {Ei}di=0 of the primitive idempotents).

A dual adjacency matrix A∗ is diagonal.

A dual adjacency matrix A∗ acts on the eigenspaces of A as follows.

A∗EiV ⊆ Ei−1V + EiV + Ei+1V (0 ≤ i ≤ d),

where E−1 = 0 and Ed+1 = 0.

An example

Example

H(2, 2) has a dual adjacency matrix

A∗ =

2 0 0 00 0 0 00 0 0 00 0 0 −2

Example

H(D, 2) has a dual adjacency matrix

A∗ =D∑i=0

(D − 2i)E ∗i .

More examples

The following graphs have a dual adjacency matrix.

Any strongly-regular graph.

Any Q-polynomial distance-regular graph, for instance:

• cycle

• Hamming graph

• Johnson graph

• Grassman graph

• Dual polar spaces

• Bilinear forms graph

• Alternating forms graph

• Hermitean forms graph

• Quadratic forms graph

See the book Distance-Regular Graphs by Brouwer, Cohen,Neumaier.

How A and A∗ are related

To summarize so far, for our graph Γ the adjacency matrix A andany dual adjacency matrix A∗ generate T . Moreover they act oneach other’s eigenspaces in the following way:

AE ∗i V ⊆ E ∗i−1V + E ∗i V + E ∗i+1V (0 ≤ i ≤ D),

where E ∗−1 = 0 and E ∗D+1 = 0.

A∗EiV ⊆ Ei−1V + EiV + Ei+1V (0 ≤ i ≤ d),

where E−1 = 0 and Ed+1 = 0.

To clarify this situation we reformulate it as a problem in linearalgebra.

Part II: Tridiagonal pairs

We now define a linear-algebraic object called a TD pair.

From now on F denotes a field.

V will denote a vector space over F with finite positive dimension.

We consider a pair of linear transformations A : V → V andA∗ : V → V .

Definition of a Tridiagonal pair (Ito, Tanabe, Ter 2001)

We say the pair A,A∗ is a TD pair on V whenever (1)–(4) holdbelow.

1 Each of A,A∗ is diagonalizable on V .

2 There exists an ordering {Vi}di=0 of theeigenspaces of A such that

A∗Vi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d),

where V−1 = 0, Vd+1 = 0.

3 There exists an ordering {V ∗i }Di=0 of the eigenspaces of A∗

such that

AV ∗i ⊆ V ∗i−1 + V ∗i + V ∗i+1 (0 ≤ i ≤ D),

where V ∗−1 = 0, V ∗D+1 = 0.

4 There is no subspace W ⊆ V such that AW ⊆W andA∗W ⊆W and W 6= 0 and W 6= V .

The diameter

Referring to our definition of a TD pair,

it turns out d = D; we call this common value the diameter of thepair.

Each irreducible T -module gives a TD pair

Briefly returning to the graph Γ, the adjacency matrix and any dualadjacency matrix act on each irreducible T -module as a TD pair.

This motivates us to understand TD pairs.

Special case: Leonard pairs

In our study of TD pairs we begin with a special case called aLeonard pair.

A Leonard pair is a TD pair for which the eigenspaces Vi and V ∗iall have dimension 1. These are classified up to isomorphism (Ter2001).

To describe the Leonard pairs, we recall some notation. A squarematrix is called tridiagonal whenever each nonzero entry lies oneither the diagonal, the subdiagonal, or the superdiagonal.

A tridiagonal matrix is irreducible whenever each entry on thesubdiagonal is nonzero and each entry on the superdiagonal isnonzero.

Description of Leonard Pairs

A pair of linear transformations A : V → V and A∗ : V → V isa Leonard pair if and only if:

1 There exists a basis for V with respect to which the matrixrepresenting A is irreducible tridiagonal and the matrixrepresenting A∗ is diagonal.

2 There exists a basis for V with respect to which the matrixrepresenting A∗ is irreducible tridiagonal and the matrixrepresenting A is diagonal.

Example of a Leonard pair

For any integer d ≥ 0 the pair

0 d 0 01 0 d − 1

2 · ·· · ·

· · 10 d 0

A∗ = diag(d , d − 2, d − 4, . . . ,−d)

is a Leonard pair on the vector space Fd+1, provided thecharacteristic of F is 0 or an odd prime greater than d .

Reason: There exists an invertible matrix P such thatP−1AP = A∗ and P2 = 2d I .

Leonard pairs and orthogonal polynomials

There is a natural correspondence between the Leonard pairs and afamily of orthogonal polynomials consisting of the following types:

q-Racah,q-Hahn,dual q-Hahn,q-Krawtchouk,dual q-Krawtchouk,quantum q-Krawtchouk,affine q-Krawtchouk,Racah,Hahn,dual-Hahn,Krawtchouk,Bannai/Ito,orphans (char(F) = 2 only).

This family coincides with the terminating branch of the Askeyscheme of orthogonal polynomials.

Recommended reading

The theory of Leonard pairs is summarized in

P. Terwilliger: An algebraic approach to the Askey scheme oforthogonal polynomials. Orthogonal polynomials and specialfunctions, 255–330, Lecture Notes in Math., 1883, Springer,Berlin, 2006; arXiv:math.QA/0408390.

Tridiagonal pairs

We now return to general TD pairs.

In 2010 my collaborators Tatsuro Ito, Kazumasa Nomura and Iclassified up to isomorphism the TD pairs over an algebraicallyclosed field.

To be precise, we classified up to isomorphism a more generalfamily of TD pairs said to be sharp.

I will describe this result shortly.

TD pairs and TD systems

When working with a TD pair, it is helpful to consider a closelyrelated object called a TD system.

We will define a TD system over the next few slides.

Standard orderings

Referring to our definition of a TD pair,

An ordering {Vi}di=0 of the eigenspaces of A is called standardwhenever

A∗Vi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d),

where V−1 = 0, Vd+1 = 0.

In this case, the ordering {Vd−i}di=0 is also standard and no furtherordering is standard.

A similar discussion applies to A∗.

Primitive idempotents

Given an eigenspace of a diagonalizable linear transformation, thecorresponding primitive idempotent E is the projection onto thateigenspace.

In other words E − I vanishes on the eigenspace and E vanishes onall the other eigenspaces.

TD systems

Definition

By a TD system on V we mean a sequence

Φ = (A; {Ei}di=0; A∗; {E ∗i }di=0)

that satisfies the following:

1 A,A∗ is a TD pair on V .

2 {Ei}di=0 is a standard ordering of the primitive idempotents ofA.

3 {E ∗i }di=0 is a standard ordering of the primitive idempotents ofA∗.

Until further notice we fix a TD system Φ as above.

The eigenvalues

For 0 ≤ i ≤ d let θi (resp. θ∗i ) denote the eigenvalue of A (resp.A∗) associated with the eigenspace EiV (resp. E ∗i V ).

We call {θi}di=0 (resp. {θ∗i }di=0) the eigenvalue sequence (resp.dual eigenvalue sequence) of Φ.

A three-term recurrence

Theorem (Ito+Tanabe+T, 2001)

The expressions

θi−2 − θi+1

θi−1 − θi,

θ∗i−2 − θ∗i+1

θ∗i−1 − θ∗i

are equal and independent of i for 2 ≤ i ≤ d − 1.

Let β + 1 denote the common value of the above expressions.

Solving the recurrence

For the above recurrence the “simplest” solution is

θi = d − 2i (0 ≤ i ≤ d),

θ∗i = d − 2i (0 ≤ i ≤ d).

In this case β = 2.

For this solution our TD system Φ is said to have Krawtchouktype.

Solving the recurrence, cont.

For the above recurrence another solution is

θi = qd−2i (0 ≤ i ≤ d),

θ∗i = qd−2i (0 ≤ i ≤ d),

q 6= 0, q2 6= 1, q2 6= −1.

In this case β = q2 + q−2.

For this solution Φ is said to have q-Krawtchouk type.

Solving the recurrence, cont.

For the above recurrence the “most general” solution is

θi = a + bq2i−d + cqd−2i (0 ≤ i ≤ d),

θ∗i = a∗ + b∗q2i−d + c∗qd−2i (0 ≤ i ≤ d),

q, a, b, c, a∗, b∗, c∗ ∈ F,q 6= 0, q2 6= 1, q2 6= −1, bb∗cc∗ 6= 0.

In this case β = q2 + q−2.

For this solution Φ is said to have q-Racah type.

Some notation

For later use we define some polynomials in an indeterminate λ.

For 0 ≤ i ≤ d ,

τi = (λ− θ0)(λ− θ1) · · · (λ− θi−1),

ηi = (λ− θd)(λ− θd−1) · · · (λ− θd−i+1),

τ∗i = (λ− θ∗0)(λ− θ∗1) · · · (λ− θ∗i−1),

η∗i = (λ− θ∗d)(λ− θ∗d−1) · · · (λ− θ∗d−i+1).

Note that each of τi , ηi , τ∗i , η∗i is monic with degree i .

The shape

It is known that for 0 ≤ i ≤ d the eigenspaces EiV , E ∗i V have thesame dimension; we denote this common dimension by ρi .

Lemma (Ito+Tanabe+T, 2001)

The sequence {ρi}di=0 is symmetric and unimodal; that is

ρi = ρd−i (0 ≤ i ≤ d),

ρi−1 ≤ ρi (1 ≤ i ≤ d/2).

We call the sequence {ρi}di=0 the shape of Φ.

A bound on the shape

Theorem (Ito+Nomura+Ter, 2009)

The shape {ρi}di=0 of Φ satisfies

ρi ≤ ρ0

)(0 ≤ i ≤ d).

The parameter ρ0

What are the possible values for ρ0?

The answer depends on the precise nature of the field F.

We will explain this after a few slides.

Some relations

Our TD system Φ = (A; {Ei}di=0; A∗; {E ∗i }di=0) satisfies thefollowing relations:

EiEj = δi ,jEi , E ∗i E ∗j = δi ,jE∗i 0 ≤ i , j ≤ d ,

1 =d∑

Ei , 1 =d∑

E ∗i ,

A =d∑

θiEi , A∗ =d∑

θ∗i E ∗i ,

E ∗i AkE ∗j = 0 if k < |i − j | 0 ≤ i , j , k ≤ d ,

EiA∗kEj = 0 if k < |i − j | 0 ≤ i , j , k ≤ d .

We call these last two equations the triple product relations.

The algebra T

Given the relations on the previous slide, it is natural to considerthe subalgebra of End(V) generated by A; A∗; {Ei}di=0; {E ∗i }di=0.

We call this algebra T .

Consider the space E ∗0 TE ∗0 .

Observe that E ∗0 TE ∗0 is an F-algebra with multiplicative identityE ∗0 .

The algebra E ∗0TE∗0

Theorem (Ito+Nomura+T, 2007)

(i) The F-algebra E ∗0 TE ∗0 is commutative and generated by

E ∗0 AiE ∗0 1 ≤ i ≤ d .

(ii) E ∗0 TE ∗0 is a field.(iii) Viewing this field as a field extension of F, the index is ρ0.

The parameter ρ0

Corollary (Ito+Nomura+Ter, 2007)

If F is algebraically closed then ρ0 = 1.

We now consider some more relations in T .

The tridiagonal relations

For our TD system Φ there exist scalars γ, γ∗, %, %∗ in F such that

A3A∗ − (β + 1)A2A∗A + (β + 1)AA∗A2 − A∗A3

= γ(A2A∗ − A∗A2) + %(AA∗ − A∗A),

A∗3A− (β + 1)A∗2AA∗ + (β + 1)A∗AA∗2 − AA∗3

= γ∗(A∗2A− AA∗2) + %∗(A∗A− AA∗).

The above equations are called the tridiagonal relations.

The Dolan-Grady relations

In the Krawtchouk case the tridiagonal relations become theDolan-Grady relations

[A, [A, [A,A∗]]] = 4[A,A∗],

[A∗, [A∗, [A∗,A]]] = 4[A∗,A].

Here [r , s] = rs − sr .

The Dolan Grady relations are the defining relations for theOnsager Algebra.

The q-Serre relations

In the q-Krawtchouk case the tridiagonal relations become thecubic q-Serre relations

A3A∗ − [3]qA2A∗A + [3]qAA∗A2 − A∗A3 = 0,

A∗3A− [3]qA∗2AA∗ + [3]qA∗AA∗2 − AA∗3 = 0.

[n]q =qn − q−n

q − q−1n = 0, 1, 2, . . .

The q-Serre relations are among the defining relations for thequantum group Uq(sl2).

The sharp case

At this point it is convenient to make an assumption about our TDsystem Φ.

Φ is called sharp whenever ρ0 = 1, where {ρi}di=0 is the shape ofΦ.

If the ground field F is algebraically closed then Φ is sharp.

Until further notice assume Φ is sharp.

The split decomposition

For 0 ≤ i ≤ d define

Ui = (E ∗0 V + · · ·+ E ∗i V ) ∩ (EiV + · · ·+ EdV ).

It is known that

V = U0 + U1 + · · ·+ Ud (direct sum),

and for 0 ≤ i ≤ d both

U0 + · · ·+ Ui = E ∗0 V + · · ·+ E ∗i V ,

Ui + · · ·+ Ud = EiV + · · ·+ EdV .

We call the sequence {Ui}di=0 the split decomposition of V withrespect to Φ.

The split decomposition, cont.

For 0 ≤ i ≤ d both

(A− θi I )Ui ⊆ Ui+1,

(A∗ − θ∗i I )Ui ⊆ Ui−1,

where U−1 = 0, Ud+1 = 0.

The split sequence, cont.

Observe that for 0 ≤ i ≤ d ,

(A− θi−1I ) · · · (A− θ1I )(A− θ0I )U0 ⊆ Ui ,

(A∗ − θ∗1I ) · · · (A∗ − θ∗i−1I )(A∗ − θ∗i I )Ui ⊆ U0.

Therefore U0 is invariant under

(A∗ − θ∗1I ) · · · (A∗ − θ∗i I )(A− θi−1I ) · · · (A− θ0I ).

Let ζi denote the corresponding eigenvalue and note that ζ0 = 1.

We call the sequence {ζi}di=0 the split sequence of Φ.

Characterizing the split sequence

The split sequence {ζi}di=0 is characterized as follows.

Lemma (Nomura+T, 2007)

For 0 ≤ i ≤ d,

E ∗0 τi (A)E ∗0 =ζiE∗0

(θ∗0 − θ∗1)(θ∗0 − θ∗2) · · · (θ∗0 − θ∗i )

A restriction on the split sequence

The split sequence {ζi}di=0 satisfies two inequalities.

Lemma (Ito+Tanabe+T, 2001)

0 6= E ∗0 EdE ∗0 ,

0 6= E ∗0 E0E ∗0 .

Consequently

0 6= ζd ,

0 6=d∑

ηd−i (θ0)η∗d−i (θ∗0)ζi .

The parameter array

Lemma (Ito+ Nomura+T, 2008)

The TD system Φ is determined up to isomorphism by thesequence

({θi}di=0; {θ∗i }di=0; {ζi}di=0).

We call this sequence the parameter array of Φ.

A classification of sharp tridiagonal systems

We are now ready to state our classification theorem for sharp TDsystems. The idea is the following.

We have seen that each sharp TD system Φ is attached to aunique parameter array

({θi}di=0; {θ∗i }di=0; {ζi}di=0). (1)

By the previous slide, this parameter array uniquely determins Φ upto isomorphism.

Therefore, to classify the sharp TD systems up to isomorphism, itsuffices to specify which sequences (1) can be their parameterarray.

This is done on the next slide.Paul Terwilliger Tridiagonal pairs in algebraic graph theory

A classification of sharp tridiagonal systems

Theorem (Ito+Nomura+Ter, 2009)

Let ({θi}di=0; {θ∗i }di=0; {ζi}di=0) (1) denote a sequence of scalarsin F. Then there exists a sharp TD system Φ over F withparameter array (1) if and only if:

(i) θi 6= θj , θ∗i 6= θ∗j if i 6= j (0 ≤ i , j ≤ d);

(ii) the expressionsθi−2−θi+1

θi−1−θi ,θ∗i−2−θ

∗i+1

θ∗i−1−θ∗i

are equal and

independent of i for 2 ≤ i ≤ d − 1;

(iii) ζ0 = 1, ζd 6= 0, and

0 6=d∑

ηd−i (θ0)η∗d−i (θ∗0)ζi .

Suppose (i)–(iii) hold. Then Φ is unique up to isomorphism of TDsystems.

The classification: proof summary

In the proof the hard part is to construct a TD system with givenparameter array of q-Racah type.

To do this we make use of the quantum affine algebra Uq(sl2).

Using the parameter array we identify two elements in Uq(sl2) thatsatisfy some tridiagonal relations.

We let these elements act on a certain Uq(sl2)-module of the formW1 ⊗W2 ⊗ · · · ⊗Wd where each Wi is an evaluation module ofdimension 2.

This action yields the desired TD system after a reduction process.

Connections

Recently TD pairs have appeared in the study of statisticalmechanics, in connection with the open XXZ spin chain. Somereferences are

P. Baseilhac, K. Koizumi. A new (in)finite dimensional algebra forquantum integrable models arXiv:math-ph/0503036

P. Baseilhac. A family of tridiagonal pairs and related symmetricfunctions. arXiv:math-ph/0604035

P. Baseilhac, K. Koizumi. A deformed analogue of Onsager’ssymmetry in the XXZ open spin chain. arXiv:hep-th/0507053

P. Baseilhac, K. Koizumi. Exact spectrum of the XXZ open spinchain from the q-Onsager algebra representation theory.arXiv:hep-th/0703106

More connections

Recently the tridiagonal relations have appeared in the study ofquantum symmetric pairs:

Stefan Kolb. Quantum symmetric Kac-Moody pairsarXiv:1207.6036.

A variation on the tridiagonal relations appears in knot theory:

Doug Bullock, Jozef H. Przytycki. Multiplicative structure ofKauffman bracket skein module quantizations Proceedings of theAMS (1999), 923–931.

A variation on the subconstituent algebra has appeared in thestudy of quantum probability:

Akihiro Hora, Nobuaki Obata. Quantum Probability and SpectralAnalysis of Graphs Series: Theoretical and Mathematical Physics2007.

Summary

In Part I we discussed the subconstituent algebra T of a graph Γand considered how the standard module V decomposes into adirect sum of irreducible T -modules.

We identified a class of graphs for which the irreducible T -modulesare nice; these graphs possess a dual adjacency matrix.

For these graphs the adjacency matrix and dual adjacency matrixact on each irreducible T -module as a TD pair.

In Part II we considered general TD pairs. We discussed theeigenvalues, dual eigenvalues, shape, tridiagonal relations,and parameter array.

We then classified up to isomorphism the TD pairs over analgebraically closed field. In the future we hope to apply thisclassification to the study of graphs.

Thank you for your attention!

THE ENDPaul Terwilliger Tridiagonal pairs in algebraic graph theory

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