Symmetric Functions and Cylindric Schur

Functions

Peter McNamara

Seminário do CAMGSD,Instituto Superior Técnico

11th October 2005

Slides and papers available from

www.math.ist.utl.pt/∼mcnamara

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 1

What is algebraic combinatorics anyway?

The biggest open problemin combinatorics:

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 2

What is algebraic combinatorics anyway?

The biggest open problemin combinatorics:

Dene combinatorics

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 2

What is algebraic combinatorics anyway?

The biggest open problemin combinatorics:

Dene combinatorics

Dene algebraic combinatorics

The use of techniques from algebra,topology,and geometry in the

solution of combinatorial problems,or the use of combinatorial

methods to attack problems in these areas.

Billera,L.J.;Björner,A.;Greene,C.;Simion,R.E.;and Stanley,R.

P.(Eds.).New Perspectives in Algebraic Combinatorics.Cambridge

University Press,1999.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 2

Thesis research

Edge labellings of

partially ordered sets

(posets)

2

60

3

12

4

20

6

10

30

15

5

1

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 3

Thesis research

Edge labellings of

partially ordered sets

(posets)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 3

Thesis research

Edge labellings of

partially ordered sets

(posets)

1

1

1

2

2

2

33

3

3

3

4

4

4

4

4

1

2

3

4

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 3

Thesis research

Edge labellings of

partially ordered sets

(posets)

1

1

1

2

2

2

33

3

3

3

4

4

4

4

4

1

2

3

4

Main Theorems

Let P be a nite graded lattice.Then the following are equiva lent:

1.

P has an S

n

EL-labelling,

2.

P is supersolvable,

3.

P has a 0-Hecke algebra action on its maximal chains,with

certain nice properties,

4.

P has a maximal chain of left-modular elements (Hugh Thomas).

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 3

Outline

Symmetric functions

Schur functions and Littlewood-Richardson coefcients

Motivation for cylindric skew Schur functions

Exposition of results

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 4

What are symmetric functions?

Denition

A

symmetric polynomial

is a polynomial that is invariant under any

permutation of its variables x

1

,x

2

,...x

n

.

Example

x

2

1

x

2

+x

2

1

x

3

+x

2

2

x

1

+x

2

2

x

3

+x

2

3

x

1

+x

2

3

x

2

is a symmetric polynomial in x

1

,x

2

,x

3

.

Denition

A

symmetric function

is a formal power series that is invariant under

any permutation of its (innite set of) variables x = (x

1

,x

2

,...).

Examples

i =j

x

2

i

x

j

is a symmetric function.

i <j

x

2

i

x

j

is

not

symmetric.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 5

Bases for the symmetric functions

Fact:

The symmetric functions form a vector space.

What is a possible basis?

Monomial symmetric functions:

Start with a monomial:

x

7

1

x

4

2

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 6

Bases for the symmetric functions

Fact:

The symmetric functions form a vector space.

What is a possible basis?

Monomial symmetric functions:

Start with a monomial:

x

7

1

x

4

2

+x

4

1

x

7

2

+x

7

1

x

4

3

+x

4

1

x

7

3

+∙ ∙ ∙.

Given a partition λ = (λ

1

,...,λ

),e.g.λ = (7,4),

m

λ

=

i

1

,...,i

distinct

x

λ

1

i

1

...x

λ

i

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 6

Bases for the symmetric functions

Fact:

The symmetric functions form a vector space.

What is a possible basis?

Monomial symmetric functions:

Start with a monomial:

x

7

1

x

4

2

+x

4

1

x

7

2

+x

7

1

x

4

3

+x

4

1

x

7

3

+∙ ∙ ∙.

Given a partition λ = (λ

1

,...,λ

),e.g.λ = (7,4),

m

λ

=

i

1

,...,i

distinct

x

λ

1

i

1

...x

λ

i

.

Elementary symmetric functions,e

λ

Complete homogeneous symmetric functions,h

λ

Power sum symmetric functions,p

λ

Typical questions:

Prove they are bases,convert from one to

another,...

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 6

Schur functions

Cauchy,1815

Partition λ = (λ

1

,λ

2

,...,λ

)

Young diagram.

Example:λ = (4,4,3,1)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 7

Schur functions

Cauchy,1815

Partition λ = (λ

1

,λ

2

,...,λ

)

Young diagram.

Example:λ = (4,4,3,1)

Semistandard Young tableau

(SSYT)

6

3

4 9

1

4

5

7

64

3 4

The Schur function s

λ

in the variables x = (x

1

,x

2

,...) is then dened

by

s

λ

=

SSYT T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

Example

s

4431

= x

1

1

x

2

3

x

4

4

x

5

x

2

6

x

7

x

9

+∙ ∙ ∙.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 7

Schur functions

Example

1 1

2

1 2

2

1 1

3

1 3

3

2 2

3

2 3

3

1 2

3

1 3

2

Hence

s

21

(x

1

,x

2

,x

3

) = x

2

1

x

2

+x

1

x

2

2

+x

2

1

x

3

+x

1

x

2

3

+x

2

2

x

3

+x

2

x

2

3

+2x

1

x

2

x

3

= m

21

(x

1

,x

2

,x

3

) +2m

111

(x

1

,x

2

,x

3

).

Fact:

Schur functions are symmetric functions.

Question

Why do we care about Schur functions?

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 8

Why do we care about Schur functions?

Fact:

The Schur functions form a basis for the symmetric

functions.

In fact,they form an orthonormal basis:s

λ

,s

µ

= δ

λµ

.

Main reason:they arise in many other areas of mathematics.

But rst...

Note:

The symmetric functions form a ring.

s

µ

s

ν

=

λ

c

λ

µν

s

λ

.

c

λ

µν

:Littlewood-Richardson coefcients

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 9

s

λ

and c

λ

µν

are superstars!

1.

Representation Theory of S

n

:

(S

µ

⊗S

ν

) ↑

S

n

=

λ

c

λ

µν

S

λ

,so χ

µ

∙ χ

ν

=

λ

c

λ

µν

χ

λ

.

We also have that

s

λ

= the Frobenius characteristic of χ

λ

.

2.

Representations of GL(n,C):

s

λ

(x

1

,...,x

n

) = the character of the irreducible rep.V

λ

.

V

µ

⊗V

ν

=

c

λ

µν

V

λ

.

3.

Algebraic Geometry:

Schubert classes σ

λ

form a linear basis for

H

∗

(Gr

kn

).We have

σ

µ

σ

ν

=

λ⊆k×(n−k)

c

λ

µν

σ

λ

.

Thus

c

λ

µν

= number of points of Gr

kn

in

˜

Ω

µ

∩

˜

Ω

ν

∩

˜

Ω

λ

∨

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 10

There's more!

4.

Linear Algebra:

When do there exist Hermitian matrices A,B

and C = A +B with eigenvalue sets µ,ν and λ,respectively?

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 11

There's more!

4.

Linear Algebra:

When do there exist Hermitian matrices A,B

and C = A +B with eigenvalue sets µ,ν and λ,respectively?

When c

λ

µν

> 0.(Heckman,Klyachko,Knutson,Tao)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 11

There's more!

4.

Linear Algebra:

When do there exist Hermitian matrices A,B

and C = A +B with eigenvalue sets µ,ν and λ,respectively?

When c

λ

µν

> 0.(Heckman,Klyachko,Knutson,Tao)

By

1

,

2

or

3

we get:

c

λ

µν

≥ 0.(Your take-home fact!)

Consequences:

We say that s

µ

s

ν

=

λ

c

λ

µν

s

λ

is a

Schur-positive

function.

Want a combinatorial proof:

They must count something simpler!

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 11

Skew Schur functions:a generalization of Schur

functions

Partition λ = (λ

1

,λ

2

,...,λ

l

)

Young diagram.

Example:

λ = (4,4,3,1)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 12

Skew Schur functions:a generalization of Schur

functions

Partition λ = (λ

1

,λ

2

,...,λ

l

)

µ ts inside λ.

Young diagram.

Example:

λ

/µ

= (4,4,3,1)

/(3,1)

4

7

5 6 6

44 9

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 12

Skew Schur functions:a generalization of Schur

functions

Partition λ = (λ

1

,λ

2

,...,λ

l

)

µ ts inside λ.

Young diagram.

Example:

λ

/µ

= (4,4,3,1)

/(3,1)

Semistandard Young tableau

(SSYT)

6

4 9

5

7

64

4

The

skew

Schur function s

λ

/µ

is the variables x = (x

1

,x

2

,...) is then

dened by

s

λ

/µ

=

SSYT T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

s

4431

/31

= x

3

4

x

5

x

2

6

x

7

x

9

+∙ ∙ ∙.Again,it's a symmetric function.

Remarkable fact:

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 12

Skew Schur functions:a generalization of Schur

functions

Partition λ = (λ

1

,λ

2

,...,λ

l

)

µ ts inside λ.

Young diagram.

Example:

λ

/µ

= (4,4,3,1)

/(3,1)

Semistandard Young tableau

(SSYT)

6

4 9

5

7

64

4

The

skew

Schur function s

λ

/µ

is the variables x = (x

1

,x

2

,...) is then

dened by

s

λ

/µ

=

SSYT T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

s

4431

/31

= x

3

4

x

5

x

2

6

x

7

x

9

+∙ ∙ ∙.Again,it's a symmetric function.

Remarkable fact:

s

λ/µ

=

ν

c

λ

µν

s

ν

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 12

The Littlewood-Richardson rule

Littlewood-Richardson 1934,Schützenberger 1977,Thomas 1974.

Theorem

c

λ

µν

equals the number of SSYT of shape λ/µ and

content

ν whose

reverse reading word

is a

ballot sequence

.

Example

λ = (5,5,2,1),µ = (3,2),ν = (4,3,1)

No

11221213 Yes

11

2 21

21

3 2

1 3

1 2 2

1 1

Yes11221312

3

1 1

2 2 2

11

11222113

to prevent bottom from getting cut off

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 13

The Littlewood-Richardson rule

Littlewood-Richardson 1934,Schützenberger 1977,Thomas 1974.

Theorem

c

λ

µν

equals the number of SSYT of shape λ/µ and

content

ν whose

reverse reading word

is a

ballot sequence

.

Example

λ = (5,5,2,1),µ = (3,2),ν = (4,3,1)

No

11221213

Yes

11

2 21

21

3 2

1 3

1 2 2

1 1

Yes

11221312

3

1 1

2 2 2

11

11222113

to prevent bottom from getting cut off

c

5521

32,431

= 2

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 13

The Littlewood-Richardson rule

Littlewood-Richardson 1934,Schützenberger 1977,Thomas 1974.

Theorem

c

λ

µν

equals the number of SSYT of shape λ/µ and

content

ν whose

reverse reading word

is a

ballot sequence

.

Example

λ = (5,5,2,1),µ = (3,2),ν = (4,3,1)

No

11221213

Yes

11

2 21

21

3 2

1 3

1 2 2

1 1

Yes

11221312

3

1 1

2 2 2

11

11222113

to prevent bottom from getting cut off

c

5521

32,431

= 2 c

(12,11,10,9,8,7,6,5,4,3,2,1)

(8,7,6,5,4,3,2,1),(8,7,6,6,5,4,3,2,1)

= 7869992

(Maple packages:John Stembridge,Anders Buch)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 13

The story so far

Schur functions:(most?) important basis for the symmetric

functions.

Skew Schur functions are Schur-positive.

The coefcients in the expansion are the Littlewood-Richar dson

coefcients c

λ

µν

.

c

λ

µν

= number of points of Gr

kn

in

˜

Ω

µ

∩

˜

Ω

ν

∩

˜

Ω

λ

∨

.

The Littlewood-Richardson rule gives a combinatorial rule for

calculating c

λ

µν

,and hence much information about the other

interpretations of c

λ

µν

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 14

Another Schur-positivity research project

Know s

µ

s

ν

=

λ

c

λ

µν

s

λ

is Schur-positive.

Question

Given µ,ν,when is

s

σ

s

τ

−s

µ

s

ν

Schur-positive?In other words,when is c

λ

στ

−c

λ

µν

≥ 0 for

every

partition λ.

Fomin,Fulton,Li,Poon:Eigenvalues,singular values,and

Littlewood-Richardson coefcients,

http://www.arxiv.org/abs/math.AG/0301307.

Bergeron,McN.:Some positive differences of products of Schur

functions, math.CO/0412289.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 15

Another Schur-positivity research project

Know s

µ

s

ν

=

λ

c

λ

µν

s

λ

is Schur-positive.

Question

Given µ,ν,when is

s

σ

s

τ

−s

µ

s

ν

Schur-positive?In other words,when is c

λ

στ

−c

λ

µν

≥ 0 for

every

partition λ.

Fomin,Fulton,Li,Poon:Eigenvalues,singular values,and

Littlewood-Richardson coefcients,

http://www.arxiv.org/abs/math.AG/0301307.

Bergeron,McN.:Some positive differences of products of Schur

functions, math.CO/0412289.

Lam,Postnikov,Pylyavskyy:Schur positivity and Schur

log-concavity math.CO/0502446.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 15

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

4

n−k

k

4

4

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

4

4

5

n−k

k

5

54

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

3

3

7 7

65

4 4

6

4 6

4

4 64

5

7

64

3 4

6

7n−k

k

54

7 7

64

6

4 6

4

Entries weakly increase in each row

Strictly increase up each column

Alternatively:SSYT with relations between entries in rst and

last columns

Cylindric skew Schur function:

s

C

(x) =

T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

e.g.s

C

(x) = x

3

x

4

4

x

5

x

3

6

x

2

7

+∙ ∙ ∙.

s

C

is a symmetric function

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

3

3

7 7

65

4 4 4

6

64

4

64

3

6

7n−k

k

754

64

7 7

65

4 4

6

4 6

4

Entries weakly increase in each row

Strictly increase up each column

Alternatively:SSYT with relations between entries in rst and

last columns

Cylindric skew Schur function:

s

C

(x) =

T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

e.g.s

C

(x) = x

3

x

4

4

x

5

x

3

6

x

2

7

+∙ ∙ ∙.

s

C

is a symmetric function

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Cylindric skew Schur functions

Innite skew shape C

Invariant under

translation

Identify (a,b) and

(a +n −k,b −k).

3

75

7

6 6

4 6

43

44

44

64

3

6

7n−k

k

75

64

75

7644

6

4 6

4

Entries weakly increase in each row

Strictly increase up each column

Alternatively:SSYT with relations between entries in rst and

last columns

Cylindric skew Schur function:

s

C

(x) =

T

x

#1's in T

1

x

#2's in T

2

∙ ∙ ∙.

e.g.s

C

(x) = x

3

x

4

4

x

5

x

3

6

x

2

7

+∙ ∙ ∙.

s

C

is a symmetric function

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 16

Skew shapes are cylindric skew shapes...

...and so skew Schur functions are cylindric skew Schur functions.

Example

k

n−k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 17

Skew shapes are cylindric skew shapes...

...and so skew Schur functions are cylindric skew Schur functions.

Example

k

n−k

Gessel,Krattenthaler:Cylindric partitions, 1997.

Bertram,Ciocan-Fontanine,Fulton:Quantummultiplication of

Schur polynomials, 1999.

Postnikov:Afne approach to quantumSchubert calculus,

math.CO/0205165.

Stanley:Recent developments in algebraic combinatorics,

math.CO/0211114.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 17

Motivation:Positivity of Gromov-Witten invariants

In H

∗

(Gr

kn

),

σ

µ

σ

ν

=

λ

c

λ

µν

σ

λ

.

In QH

∗

(Gr

kn

),

σ

µ

∗ σ

ν

=

d≥0

λ⊆k×(n−k)

q

d

C

λ,d

µν

σ

λ

.

C

λ,d

µν

= 3-point

Gromov-Witten invariants

=#{rational curves of degree d in Gr

kn

that meet

˜

Ω

µ

,

˜

Ω

ν

and

˜

Ω

λ

∨

}.

Example

C

λ,

0

µ,ν

= c

λ

µν

.

Key point:

C

λ,d

µν

≥ 0.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 18

Motivation:Positivity of Gromov-Witten invariants

In H

∗

(Gr

kn

),

σ

µ

σ

ν

=

λ

c

λ

µν

σ

λ

.

In QH

∗

(Gr

kn

),

σ

µ

∗ σ

ν

=

d≥0

λ⊆k×(n−k)

q

d

C

λ,d

µν

σ

λ

.

C

λ,d

µν

= 3-point

Gromov-Witten invariants

=#{rational curves of degree d in Gr

kn

that meet

˜

Ω

µ

,

˜

Ω

ν

and

˜

Ω

λ

∨

}.

Example

C

λ,

0

µ,ν

= c

λ

µν

.

Key point:

C

λ,d

µν

≥ 0.

Fundamental open problem:

Find an algebraic or combinatorial

proof of this fact.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 18

Connection with cylindric skew Schur functions

Theorem (Postnikov)

s

µ/d/ν

(x

1

,...,x

k

) =

λ⊆k×(n−k)

C

λ,d

µν

s

λ

(x

1

,...,x

k

).

Conclusion:

Want to understand the expansions of cylindric skew

Schur functions into Schur functions.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 19

Connection with cylindric skew Schur functions

Theorem (Postnikov)

s

µ/d/ν

(x

1

,...,x

k

) =

λ⊆k×(n−k)

C

λ,d

µν

s

λ

(x

1

,...,x

k

).

Conclusion:

Want to understand the expansions of cylindric skew

Schur functions into Schur functions.

Corollary

s

µ/d/ν

(x

1

,...,x

k

) is Schur-positive.

Known:

s

µ/d/ν

(x

1

,x

2

,...) ≡ s

µ/d/ν

(x) need not be Schur-positive.

Example

If s

µ/d/ν

= s

22

+s

211

−s

1111

,then s

µ/d/ν

(x

1

,x

2

,x

3

) is Schur-positive.

(In general:s

λ

(x

1

,...,x

k

) = 0 ⇔λ has at most k parts.)

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 19

When is a cylindric skew Schur function

Schur-positive?

k

n−k

Theorem (McN.)

For any cylindric skew shape C,

s

C

(x

1

,x

2

,...) is Schur-positive ⇔C is a skew shape.

Equivalently,if C is a non-trivial cylindric skew shape,then

s

C

(x

1

,x

2

,...) is

not

Schur-positive.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 20

Example:cylindric ribbons

C:

k

n−k

s

C

(x

1

,x

2

,...) =

λ⊆k×(n−k)

c

λ

s

λ

+s

(n−k,1

k

)

−s

(n−k−1,1

k+1

)

+s

(n−k−2,1

k+2

)

−∙ ∙ ∙ +(−1)

n−k

s

(1

n

)

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 21

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

+

+

=

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

=

++

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

+

= +

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

++

=

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

++

=

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

++=

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

Formula:cylindric skew Schur functions as signed

sums of skew Schur functions

Idea for formulation:Bertram,Ciocan-Fontanine,Fulton

Uses result of Gessel,Krattenthaler

Example

++=

n−k

k

s

C

= s

333211/21

−s

3322111/21

+s

331111111/21

=

s

3331

+s

3322

+s

33211

+s

322111

+s

31111111

−s

222211

−s

2221111

+s

22111111

+s

211111111

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 22

First consequence:lots of skew Schur function

identities

+

+=

+

+=

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 23

A nal thought:shouldn't cylindric skew Schur

functions be Schur-positive in some sense?

C:

H:

k

k

n−k

n−k

s

C

(x

1

,x

2

,...) =

λ⊆k×(n−k)

c

λ

s

λ

+s

(n−k,1

k

)

−s

(n−k−1,1

k+1

)

+s

(n−k−2,1

k+2

)

−∙ ∙ ∙ +(−1)

n−k

s

(1

n

)

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 24

A nal thought:shouldn't cylindric skew Schur

functions be Schur-positive in some sense?

C:

H:

k

n−k

k

n−k

s

C

(x

1

,x

2

,...) =

λ⊆k×(n−k)

c

λ

s

λ

+s

(n−k,1

k

)

−s

(n−k−1,1

k+1

)

+s

(n−k−2,1

k+2

)

−∙ ∙ ∙ +(−1)

n−k

s

(1

n

)

.

In fact,

s

C

(x

1

,x

2

,...) =

λ⊆k×(n−k)

c

λ

s

λ

+s

H

.

s

C

:cylindric skew Schur function

s

H

:cylindric Schur function

We say that s

C

is

cylindric Schur-positive

.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 24

A Conjecture

=

+

n−k

k

n−k

k

n−k

k

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 25

A Conjecture

=

+

n−k

k

n−k

k

n−k

k

Conjecture

For any cylindric skew shape C,s

C

is cylindric Schur-positive

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 25

A Conjecture

=

+

n−k

k

n−k

k

n−k

k

Conjecture

For any cylindric skew shape C,s

C

is cylindric Schur-positive

Theorem (McN.)

The cylindric Schur functions corresponding to a given translation

(−n +k,+k) are linearly independent.

Theorem (McN.)

If s

C

can be written as a linear combination of cylindric Schur

functions with the same translation as C,then s

C

is cylindric

Schur-positive.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 25

Summary of results

Classication of those cylindric skew Schur functions that are

Schur-positive.

Full knowledge of negative terms in Schur expansion of ribbons.

Expansion of any cylindric skew Schur function into skew Schur

functions.

Conjecture and evidence that every cylindric skew Schur

function is cylindric Schur-positive.

Outlook

Prove the conjecture.

Develop a Littlewood-Richardson rule for cylindric skew Schur

functions - this would solve the fundamental open problem.

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 26

The Schur function s

λ

is a symmetric function

Proof.

Consider SSYTs of shape λ and content α = (α

1

,α

2

,...).

Show:

#SSYTs shape λ,content α =#SSYTs shape λ,content β,

where β is any permutation of α.

Sufcient:

β = (α

1

,...,α

i −1

,

α

i +1

,

α

i

,α

i +2

,...).

Bijection:

SSYTs shape λ,content α ↔SSYTs shape λ,content β.

i +1 i +1

i i i i

i +1 i +1 i +1 i +1

i +1

r=2 s=4

i

In each such row,convert the r i's and s i +1's to s i's and r i +1's:

i +1 i +1

i i i i i i

i +1 i +1

i +1

s=4 r=2

i

Symmetric Functions and Cylindric Schur Functions

Peter McNamara 27

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