ocw presentation

A Schur-Like Basis of NSym Definedby a Pieri Rule

John M. Campbell

Joint work with Karen Feldman, Jennifer Light, PavelSchuldiner, and Yan Xu

ש! + ש! + ש!

NSym, QSym, and Sym

NSym denotes the algebra of non-commutative symmetricfunctions.

QSym denotes the algebra of quasi-symmetric functions.Sym denotes the algebra of symmetric functions.

NSym = Q〈H1,H2, · · · 〉

QSym is a subalgebra of Q[[x1, x2, . . . ]] with monomialquasi-symmetric functions

Mα =∑

i1<i2<···<im

xα1i1

xα2i2· · · xαm

im

forming a basis.

Sym = Q[h1, h2, · · · ]

NSym, QSym, and Sym

NSym denotes the algebra of non-commutative symmetricfunctions.QSym denotes the algebra of quasi-symmetric functions.

Sym denotes the algebra of symmetric functions.

NSym = Q〈H1,H2, · · · 〉


Mα =∑

i1<i2<···<im

xα1i1

xα2i2· · · xαm

im

forming a basis.

Sym = Q[h1, h2, · · · ]

NSym, QSym, and Sym

NSym denotes the algebra of non-commutative symmetricfunctions.QSym denotes the algebra of quasi-symmetric functions.Sym denotes the algebra of symmetric functions.

NSym = Q〈H1,H2, · · · 〉


Mα =∑

i1<i2<···<im

xα1i1

xα2i2· · · xαm

im

forming a basis.

Sym = Q[h1, h2, · · · ]

NSym, QSym, and Sym

Recall that NSym is generated by {H1,H2, · · · } over Q.

To multiply elements in the canonical generating set of NSym,simply concatenate the indices to form a composition as illustratedbelow.

H H H = H

NSym, QSym, and Sym

Recall that NSym is generated by {H1,H2, · · · } over Q.To multiply elements in the canonical generating set of NSym,simply concatenate the indices to form a composition as illustratedbelow.

H H H = H

NSym, QSym, and Sym

Recall that Sym is generated by {h1, h2, · · · } over Q, andhihj = hjhi for all indices i and j .

To multiply elements in this generating set, concatenate the indicesand then sort the indices to form a partition as illustrated below.

h h h = h

NSym, QSym, and Sym

Recall that Sym is generated by {h1, h2, · · · } over Q, andhihj = hjhi for all indices i and j .To multiply elements in this generating set, concatenate the indicesand then sort the indices to form a partition as illustrated below.

h h h = h

NSym, QSym, and Sym

NSym, QSym, and Sym are graded combinatorial Hopfalgebras.

Bases of NSym and QSym are indexed by compositions.Bases of Sym are indexed by partitions.{Hα}α∈C is the complete homogeneous basis of NSym, and{hλ}λ∈P is the complete homogeneous basis of Sym.Let NSymr denote the (additive abelian) group of all homogeneouspolynomials in NSym of degree r together with 0.

NSym =⊕r≥0

NSymr

Let Symr denote the (additive abelian) group of all homogeneouspolynomials in Sym of degree r together with 0.

Sym =⊕r≥0

Symr

NSym, QSym, and Sym

NSym, QSym, and Sym are graded combinatorial Hopfalgebras.Bases of NSym and QSym are indexed by compositions.

Bases of Sym are indexed by partitions.{Hα}α∈C is the complete homogeneous basis of NSym, and{hλ}λ∈P is the complete homogeneous basis of Sym.Let NSymr denote the (additive abelian) group of all homogeneouspolynomials in NSym of degree r together with 0.

NSym =⊕r≥0

NSymr


Sym =⊕r≥0

Symr

NSym, QSym, and Sym

NSym, QSym, and Sym are graded combinatorial Hopfalgebras.Bases of NSym and QSym are indexed by compositions.Bases of Sym are indexed by partitions.

{Hα}α∈C is the complete homogeneous basis of NSym, and{hλ}λ∈P is the complete homogeneous basis of Sym.Let NSymr denote the (additive abelian) group of all homogeneouspolynomials in NSym of degree r together with 0.

NSym =⊕r≥0

NSymr


Sym =⊕r≥0

Symr

NSym, QSym, and Sym

NSym, QSym, and Sym are graded combinatorial Hopfalgebras.Bases of NSym and QSym are indexed by compositions.Bases of Sym are indexed by partitions.{Hα}α∈C is the complete homogeneous basis of NSym, and{hλ}λ∈P is the complete homogeneous basis of Sym.

Let NSymr denote the (additive abelian) group of all homogeneouspolynomials in NSym of degree r together with 0.

NSym =⊕r≥0

NSymr


Sym =⊕r≥0

Symr

NSym, QSym, and Sym

NSym, QSym, and Sym are graded combinatorial Hopfalgebras.Bases of NSym and QSym are indexed by compositions.Bases of Sym are indexed by partitions.{Hα}α∈C is the complete homogeneous basis of NSym, and{hλ}λ∈P is the complete homogeneous basis of Sym.Let NSymr denote the (additive abelian) group of all homogeneouspolynomials in NSym of degree r together with 0.

NSym =⊕r≥0

NSymr


Sym =⊕r≥0

Symr

The Jacobi-Trudi Formula

The Schur basis {sλ}λ∈P of Sym is an important basis of Sym.

Let λ = (λ1, λ2, · · · , λn) be a partition.The Schur function sλ may be defined using the Jacobi-Trudiformula:

sλ = det

hλ1 hλ1+1 · · · hλ1+n−1

hλ2−1 hλ2 · · · hλ2+n−2...

.... . .

...hλn−n+1 hλn−n+2 · · · hλn

= det[hλi+j−i ]1≤i ,j≤n


The Schur basis {sλ}λ∈P of Sym is an important basis of Sym.Let λ = (λ1, λ2, · · · , λn) be a partition.

The Schur function sλ may be defined using the Jacobi-Trudiformula:

sλ = det

hλ1 hλ1+1 · · · hλ1+n−1

hλ2−1 hλ2 · · · hλ2+n−2...

.... . .

...hλn−n+1 hλn−n+2 · · · hλn



The Schur basis {sλ}λ∈P of Sym is an important basis of Sym.Let λ = (λ1, λ2, · · · , λn) be a partition.The Schur function sλ may be defined using the Jacobi-Trudiformula:

sλ = det

hλ1 hλ1+1 · · · hλ1+n−1

hλ2−1 hλ2 · · · hλ2+n−2...

.... . .

...hλn−n+1 hλn−n+2 · · · hλn



For example, the Schur function s(4,3,1) ∈ Sym may be evaluatedas follows using the Jacobi-Trudi formula:

s(4,3,1) =

∣∣∣∣∣∣h4 h5 h6

h2 h3 h4

0 1 h1

∣∣∣∣∣∣3×3

= h(4,3,1) − h(4,4) − h(5,2,1) + h(6,2)

The Jacobi-Trudi formula may be interpreted combinatorially usinglattice paths

The Schur-Pieri Rule

One may define the Schur basis combinatorially using theSchur-Pieri rule.

Let µ and λ be partitions. Then µ differs from λ by a horizontalstrip if µ contains λ and in a given column of the diagram of µ,there is at most one cell in the diagram of µ which is not in thediagram of λ.

Theorem (The Schur-Pieri Rule)

sλhr =∑

µ`|λ|+r sµ, where the sum is over all partitions µ ` |λ|+ rwhich differ from λ by a horizontal strip of size r .


One may define the Schur basis combinatorially using theSchur-Pieri rule.Let µ and λ be partitions. Then µ differs from λ by a horizontalstrip if µ contains λ and in a given column of the diagram of µ,there is at most one cell in the diagram of µ which is not in thediagram of λ.

Theorem (The Schur-Pieri Rule)

sλhr =∑

µ`|λ|+r sµ, where the sum is over all partitions µ ` |λ|+ rwhich differ from λ by a horizontal strip of size r .


s h2 =

s + s + s + s

Young Tableaux

The Schur-Pieri rule is related to combinatorial objects calledYoung tableaux.

Let λ and µ be partitions. Then a column strict (or semi standard)Young tableau of shape λ and content µ is a tableau of shape λwith µ1 1’s, µ2 2’s, and so forth, such that the rows are weaklyincreasing and the columns are strictly increasing.For example, there are 4 column strict Young tableaux of shape(5, 3, 1) and content (3, 1, 4, 1).

43 3 31 1 1 2 3

42 3 31 1 1 3 3

32 3 41 1 1 3 3

32 3 31 1 1 3 4

Young Tableaux

The Schur-Pieri rule is related to combinatorial objects calledYoung tableaux.Let λ and µ be partitions. Then a column strict (or semi standard)Young tableau of shape λ and content µ is a tableau of shape λwith µ1 1’s, µ2 2’s, and so forth, such that the rows are weaklyincreasing and the columns are strictly increasing.

For example, there are 4 column strict Young tableaux of shape(5, 3, 1) and content (3, 1, 4, 1).

43 3 31 1 1 2 3

42 3 31 1 1 3 3

32 3 41 1 1 3 3

32 3 31 1 1 3 4

Young Tableaux

The Schur-Pieri rule is related to combinatorial objects calledYoung tableaux.Let λ and µ be partitions. Then a column strict (or semi standard)Young tableau of shape λ and content µ is a tableau of shape λwith µ1 1’s, µ2 2’s, and so forth, such that the rows are weaklyincreasing and the columns are strictly increasing.For example, there are 4 column strict Young tableaux of shape(5, 3, 1) and content (3, 1, 4, 1).

43 3 31 1 1 2 3

42 3 31 1 1 3 3

32 3 41 1 1 3 3

32 3 31 1 1 3 4

Kostka Numbers

The Kostka number Kλµ is the number of column strict Youngtableaux of shape λ and content µ.

The following formula follows from a repeated application of theSchur-Pieri rule.

hµ =∑λ≥`µ

Kλ,µsλ.

Kostka Numbers

The Kostka number Kλµ is the number of column strict Youngtableaux of shape λ and content µ.The following formula follows from a repeated application of theSchur-Pieri rule.

hµ =∑λ≥`µ

Kλ,µsλ.

Main Problems

Here is the main problems we will be discussing in thispresentation:

Is there a natural non-commutative analogue of theSchur basis, which projects onto the Schur basis in anatural way?

This is an important problems in algebraic combinatorics.

The Immaculate Basis of NSym

In 2012, Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano,and Mike Zabrocki introduced a new basis of NSym, theimmaculate basis {Sα}α∈C of NSym.

Schur functions may be defined using Bernstein operators. Theimmaculate basis may be defined using a non-commutativeanalogue of Bernstein operators.The immaculate basis satisfies a non-commutative analogue of theJacobi-Trudi formula, and may be defined using this formula.The immaculate basis also satisfies a non-commutative analogue ofthe Schur-Pieri rule, and may be defined using this analogue of theclassical Pieri rule.This immaculate-Pieri rule may be defined using anon-commutative analogue of horizontal strips.


In 2012, Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano,and Mike Zabrocki introduced a new basis of NSym, theimmaculate basis {Sα}α∈C of NSym.Schur functions may be defined using Bernstein operators. Theimmaculate basis may be defined using a non-commutativeanalogue of Bernstein operators.

The immaculate basis satisfies a non-commutative analogue of theJacobi-Trudi formula, and may be defined using this formula.The immaculate basis also satisfies a non-commutative analogue ofthe Schur-Pieri rule, and may be defined using this analogue of theclassical Pieri rule.This immaculate-Pieri rule may be defined using anon-commutative analogue of horizontal strips.


In 2012, Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano,and Mike Zabrocki introduced a new basis of NSym, theimmaculate basis {Sα}α∈C of NSym.Schur functions may be defined using Bernstein operators. Theimmaculate basis may be defined using a non-commutativeanalogue of Bernstein operators.The immaculate basis satisfies a non-commutative analogue of theJacobi-Trudi formula, and may be defined using this formula.

The immaculate basis also satisfies a non-commutative analogue ofthe Schur-Pieri rule, and may be defined using this analogue of theclassical Pieri rule.This immaculate-Pieri rule may be defined using anon-commutative analogue of horizontal strips.


In 2012, Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano,and Mike Zabrocki introduced a new basis of NSym, theimmaculate basis {Sα}α∈C of NSym.Schur functions may be defined using Bernstein operators. Theimmaculate basis may be defined using a non-commutativeanalogue of Bernstein operators.The immaculate basis satisfies a non-commutative analogue of theJacobi-Trudi formula, and may be defined using this formula.The immaculate basis also satisfies a non-commutative analogue ofthe Schur-Pieri rule, and may be defined using this analogue of theclassical Pieri rule.

This immaculate-Pieri rule may be defined using anon-commutative analogue of horizontal strips.


In 2012, Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano,and Mike Zabrocki introduced a new basis of NSym, theimmaculate basis {Sα}α∈C of NSym.Schur functions may be defined using Bernstein operators. Theimmaculate basis may be defined using a non-commutativeanalogue of Bernstein operators.The immaculate basis satisfies a non-commutative analogue of theJacobi-Trudi formula, and may be defined using this formula.The immaculate basis also satisfies a non-commutative analogue ofthe Schur-Pieri rule, and may be defined using this analogue of theclassical Pieri rule.This immaculate-Pieri rule may be defined using anon-commutative analogue of horizontal strips.

Non-Commutative Analogues of the Schur Basis

The tableaux resulting from a repeated application of theimmaculate-Pieri rule (immaculate tableaux) are onlycolumn-strict in the first column.

Although immaculate tableaux are interesting combinatorialobjects in their own right, it seems that a non-commutativeanalogue of a column-strict Young tableau should be column strictin each column.

Non-Commutative Analogues of the Schur Basis

The tableaux resulting from a repeated application of theimmaculate-Pieri rule (immaculate tableaux) are onlycolumn-strict in the first column.Although immaculate tableaux are interesting combinatorialobjects in their own right, it seems that a non-commutativeanalogue of a column-strict Young tableau should be column strictin each column.

Column-Strict Tableaux

Let α and β be compositions. We define a tableau-ש! (pronounced“shin-tableau”) of shape α and content β to be a tableau of shapeα with β1 1’s, β2 2’s, and so forth, such that the columns arestrictly increasing and the rows are weakley increasing.

tableaux-ש! seem to be a natural analogue of column-strict Youngtableaux.There are two tableaux-ש! of shape (5, 1, 3) and content (3, 1, 4, 1):

3 3 421 1 1 3 3

3 3 321 1 1 3 4

.


Let α and β be compositions. We define a tableau-ש! (pronounced“shin-tableau”) of shape α and content β to be a tableau of shapeα with β1 1’s, β2 2’s, and so forth, such that the columns arestrictly increasing and the rows are weakley increasing.tableaux-ש! seem to be a natural analogue of column-strict Youngtableaux.

There are two tableaux-ש! of shape (5, 1, 3) and content (3, 1, 4, 1):

3 3 421 1 1 3 3

3 3 321 1 1 3 4

.


Let α and β be compositions. We define a tableau-ש! (pronounced“shin-tableau”) of shape α and content β to be a tableau of shapeα with β1 1’s, β2 2’s, and so forth, such that the columns arestrictly increasing and the rows are weakley increasing.tableaux-ש! seem to be a natural analogue of column-strict Youngtableaux.There are two tableaux-ש! of shape (5, 1, 3) and content (3, 1, 4, 1):

3 3 421 1 1 3 3

3 3 321 1 1 3 4

.

The basis-ש! of NSym

The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).

The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.

The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.

The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.

The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.

tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.

We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


The basis-ש! (which will be defined later) was first uncovered bycomputational experiments of Chris Berg (2012).The basis-ש! projects in a natural way onto the Schur basis.The basis-ש! has a number of Schur-like properties.The basis-ש! is defined using a Pieri rule. The tableaux resultingfrom this Pieri rule are ,tableaux-ש! i.e. column-strict in eachcolumn.The main advantage of the basis-ש! over the immaculate basis isthat tableaux-ש! seem to be better analogues of column-strictYoung tableaux compared to immaculate tableaux.tableaux-ש! are interesting combinatorial objects in their own right.We have discovered a variety of combinatorial formulas involvingthe .basis-ש!


We begin by constructing a non-commutative analogue ofhorizontal strips.

Let α and β be compositions. Then β differs from α by ashin-horizontal strip of size r iff: α is contained in β,|β| = |α|+ r and for all 1 ≤ i ≤ `(α), if βi > αi , then for all j > i ,βj ≤ αi .The shin function ש! : C → NSym maps an arbitrary compositionα to the unique non-commutative symmetric function αש! satisfyingαHrש! =

∑β ,βש! where the sum is over all compositions β which

differ from α by a shin-horizontal strip of size r .


We begin by constructing a non-commutative analogue ofhorizontal strips.Let α and β be compositions. Then β differs from α by ashin-horizontal strip of size r iff: α is contained in β,|β| = |α|+ r and for all 1 ≤ i ≤ `(α), if βi > αi , then for all j > i ,βj ≤ αi .

The shin function ש! : C → NSym maps an arbitrary compositionα to the unique non-commutative symmetric function αש! satisfyingαHrש! =




We begin by constructing a non-commutative analogue ofhorizontal strips.Let α and β be compositions. Then β differs from α by ashin-horizontal strip of size r iff: α is contained in β,|β| = |α|+ r and for all 1 ≤ i ≤ `(α), if βi > αi , then for all j > i ,βj ≤ αi .The shin function ש! : C → NSym maps an arbitrary compositionα to the unique non-commutative symmetric function αש! satisfyingαHrש! =



The Pieri-ש! rule

ש! H2 =

ש! + ש! + ש!

α∈C{αש!} is a basis of NSym

One may use a combinatorial argument to prove that { α}α∈Cש! is abasis of NSym.

By a repeated application of the Pieri-ש! rule, it is not difficult toprove that the shin function has a positive, uni-triangularexpansion in the complete homogeneous basis of NSym.This argument involves the enumeration of .tableaux-ש!{ α}α∈Cש! is the basis-ש! of NSym. Elements of the basis are.functions-ש!


One may use a combinatorial argument to prove that { α}α∈Cש! is abasis of NSym.By a repeated application of the Pieri-ש! rule, it is not difficult toprove that the shin function has a positive, uni-triangularexpansion in the complete homogeneous basis of NSym.

This argument involves the enumeration of .tableaux-ש!{ α}α∈Cש! is the basis-ש! of NSym. Elements of the basis are.functions-ש!


One may use a combinatorial argument to prove that { α}α∈Cש! is abasis of NSym.By a repeated application of the Pieri-ש! rule, it is not difficult toprove that the shin function has a positive, uni-triangularexpansion in the complete homogeneous basis of NSym.This argument involves the enumeration of .tableaux-ש!

{ α}α∈Cש! is the basis-ש! of NSym. Elements of the basis are.functions-ש!


One may use a combinatorial argument to prove that { α}α∈Cש! is abasis of NSym.By a repeated application of the Pieri-ש! rule, it is not difficult toprove that the shin function has a positive, uni-triangularexpansion in the complete homogeneous basis of NSym.This argument involves the enumeration of .tableaux-ש!{ α}α∈Cש! is the basis-ש! of NSym. Elements of the basis are.functions-ש!

The Forgetful Map χ : NSym→ Sym

Define the linear map χ : NSym→ Sym so that:

χ (Hα) = hα1hα2 · · · hα`(α) .

χ is a surjective Hopf algebra homomorphism.χ is called a projection morphism or simply a projection.χ is also called a forgetful map because it “forgets” thatmultiplication in NSym is non-commutative.For example,

χ(H(3,1,4)) = h3h1h4 = h(4,3,1)

χ

(3

2H(2,2) −

1

7H(1,3)

)=

3

2h(2,2) −

1

7h(3,1)



χ (Hα) = hα1hα2 · · · hα`(α) .

χ is a surjective Hopf algebra homomorphism.

χ is called a projection morphism or simply a projection.χ is also called a forgetful map because it “forgets” thatmultiplication in NSym is non-commutative.For example,

χ(H(3,1,4)) = h3h1h4 = h(4,3,1)

χ

(3

2H(2,2) −

1

7H(1,3)

)=

3

2h(2,2) −

1

7h(3,1)



χ (Hα) = hα1hα2 · · · hα`(α) .

χ is a surjective Hopf algebra homomorphism.χ is called a projection morphism or simply a projection.

χ is also called a forgetful map because it “forgets” thatmultiplication in NSym is non-commutative.For example,

χ(H(3,1,4)) = h3h1h4 = h(4,3,1)

χ

(3

2H(2,2) −

1

7H(1,3)

)=

3

2h(2,2) −

1

7h(3,1)



χ (Hα) = hα1hα2 · · · hα`(α) .

χ is a surjective Hopf algebra homomorphism.χ is called a projection morphism or simply a projection.χ is also called a forgetful map because it “forgets” thatmultiplication in NSym is non-commutative.

For example,

χ(H(3,1,4)) = h3h1h4 = h(4,3,1)

χ

(3

2H(2,2) −

1

7H(1,3)

)=

3

2h(2,2) −

1

7h(3,1)



χ (Hα) = hα1hα2 · · · hα`(α) .

χ is a surjective Hopf algebra homomorphism.χ is called a projection morphism or simply a projection.χ is also called a forgetful map because it “forgets” thatmultiplication in NSym is non-commutative.For example,

χ(H(3,1,4)) = h3h1h4 = h(4,3,1)

χ

(3

2H(2,2) −

1

7H(1,3)

)=

3

2h(2,2) −

1

7h(3,1)

A Projection Formula

Through computational experiments, Chris Berg observed that thecommutative image of λש! is sλ for a partition λ, and χ(!שα) = 0 fora non-partition composition α.

Theorem

For a composition α,

χ(!שα) =

{sα if α is a partition

0 otherwise.


χ( (αש! =


0 otherwise

We initially proved this theorem using poset induction.Mike Zabrocki proved this theorem using the duality of NSym andQSym.


χ( (αש! =


0 otherwise

We initially proved this theorem using poset induction.

Mike Zabrocki proved this theorem using the duality of NSym andQSym.


χ( (αש! =


0 otherwise

We initially proved this theorem using poset induction.Mike Zabrocki proved this theorem using the duality of NSym andQSym.

Expressions of the Form αRβש!

One of our main results involving the shin basis is a theorem forevaluating expressions of the form αRβש! in the ,basis-ש! where Rβis the element of the ribbon basis indexed by β.

This theorem involves a variety of combinatorial objects, such asribbons, standard skew shin-tableaux, and descent sets.

Theorem (2013)

For all α, β ∈ C,

αRβש! =∑T

γש! ,

where the sum is over all standard skew shin-tableaux T of innershape γ/α such that D(T ) = D(β), where γ � |α|+ |β|.

Expressions of the Form αRβש!

One of our main results involving the shin basis is a theorem forevaluating expressions of the form αRβש! in the ,basis-ש! where Rβis the element of the ribbon basis indexed by β.This theorem involves a variety of combinatorial objects, such asribbons, standard skew shin-tableaux, and descent sets.

Theorem (2013)

For all α, β ∈ C,

αRβש! =∑T

γש! ,

where the sum is over all standard skew shin-tableaux T of innershape γ/α such that D(T ) = D(β), where γ � |α|+ |β|.

Hooks-ש!

This theorem turns out to be unexpectedly useful.

For example, this theorem allows us to evaluate shin functionsindexed by hooks in the complete homogeneous basis.

Corollary

For an arbitrary composition of the form (n, 1m),

(n,1m)ש! = R(n,1m) =∑

α�n+m,α1≥n(−1)m+1−`(α)Hα .

Hooks-ש!

This theorem turns out to be unexpectedly useful.For example, this theorem allows us to evaluate shin functionsindexed by hooks in the complete homogeneous basis.

Corollary

For an arbitrary composition of the form (n, 1m),

(n,1m)ש! = R(n,1m) =∑

α�n+m,α1≥n(−1)m+1−`(α)Hα .

Rectangles-ש!

Computational experiments suggest that this ribbon multiplicationformula may be used to construct a sign-reversing involution toprove the below proposition.

Proposition (2014)

For n ∈ N,2nש! =

∑α

(−1)# of 1’s in αRα,

where the sum is over all compositions α � 2n of length n suchthat α1 ≥ 2, αn ≤ 2, and for i ,m ∈ N, αi = m > 2 if and only if

αi+1 = αi+2 = · · · = αi+m−2 = 1,

adopting the convention that αi = 0 for i > `(α).

Rectangles-ש!

ש! = R − R − R + R

−R + R + R − R

A Murnaghan-Nakayama rule for the basis-ש!

The basis-ש! satisfies a number of Schur-like properties.

For example, the basis-ש! satisfies a non-commutative analogue ofthe classical Murnaghan-Nakayama rule.

Theorem (2013)

Letting α ∈ C and n ∈ N be arbitrary,

αψnש! =∑β

(−1)height(β/α)−1 ,βש!

where the sum is over all compositions β such that β/α is a.slinky-ש!

A Murnaghan-Nakayama rule for the basis-ש!

The basis-ש! satisfies a number of Schur-like properties.For example, the basis-ש! satisfies a non-commutative analogue ofthe classical Murnaghan-Nakayama rule.

Theorem (2013)

Letting α ∈ C and n ∈ N be arbitrary,

αψnש! =∑β

(−1)height(β/α)−1 ,βש!

where the sum is over all compositions β such that β/α is a.slinky-ש!

“Diving Boards” and Reverse Hooks

Theorem (2013)

For m > 1,

1n,m,1r)ש! ) =r∑

i=0

n∑j=0

(−1)i+jEn−jHm+iHjEr−i

Corollary

For m > 1,

(1n,m)ש! =n∑

j=0

(−1)jEn−jH(m,j)

Remark: There is no known formula for expressions of the formS(1n,m,1r ), or expressions of the form S(1n,m).

“Diving Boards” and Reverse Hooks

We proved the formula

1n,m,1r)ש! ) =r∑

i=0

n∑j=0

(−1)i+jEn−jHm+iHjEr−i

using induction, but we also constructed an elegant combinatorialproof involving skew-shin tableaux of the reverse hook formula

(1n,m)ש! =n∑

j=0

(−1)jEn−jH(m,j)

using a sign-reversing involution.

Summary

1 The basis-ש! projects in a natural way onto the Schur basis.

2 tableaux-ש! are arguably better analogues of column-strictYoung tableaux compared to immaculate tableaux.

3 There is a simple combinatorial formula which relates thebasis-ש! and the ribbon basis.

4 The basis-ש! satisfies a Murnaghan-Nakayama-like rule.

5 Elements of the basis-ש! indexed by certain kinds ofcompositions, such as hooks, reverse hooks, diving boards,and rectangles, seem to be interesting combinatorial objects.

ocw presentation

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