Symmetric Functions and Cylindric Schur Functions

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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:
Dene combinatorics
Symmetric Functions and Cylindric Schur Functions
Peter McNamara 2
What is algebraic combinatorics anyway?
The biggest open problemin combinatorics:
Dene combinatorics
Dene 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 coefcients
￿
Motivation for cylindric skew Schur functions
￿
Exposition of results
Symmetric Functions and Cylindric Schur Functions
Peter McNamara 4
What are symmetric functions?
Denition
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
.
Denition
A
symmetric function
is a formal power series that is invariant under
any permutation of its (innite 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

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

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 dened
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 coefcients
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

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

2
,...,λ
l
)
￿
µ ts inside λ.
￿
Young diagram.
Example:
λ

= (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

2
,...,λ
l
)
￿
µ ts inside λ.
￿
Young diagram.
Example:
λ

= (4,4,3,1)
/(3,1)
￿
Semistandard Young tableau
(SSYT)
6
4 9
5
7
64
4
The
skew
Schur function s
λ

is the variables x = (x
1
,x
2
,...) is then
dened by
s
λ

=
￿
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

2
,...,λ
l
)
￿
µ ts inside λ.
￿
Young diagram.
Example:
λ

= (4,4,3,1)
/(3,1)
￿
Semistandard Young tableau
(SSYT)
6
4 9
5
7
64
4
The
skew
Schur function s
λ

is the variables x = (x
1
,x
2
,...) is then
dened by
s
λ

=
￿
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 coefcients in the expansion are the Littlewood-Richar dson
coefcients 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 coefcients,
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 coefcients,
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
￿
Innite 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
￿
Innite 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
￿
Innite 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
￿
Innite 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
￿
Innite 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
￿
Innite 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:Afne 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
￿
Classication 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

2
,...).
Show:
#SSYTs shape λ,content α =#SSYTs shape λ,content β,
where β is any permutation of α.
Sufcient:
β = (α
1
,...,α
i −1
,
α
i +1
,
α
i

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