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
ELlabelling,
2.
P is supersolvable,
3.
P has a 0Hecke algebra action on its maximal chains,with
certain nice properties,
4.
P has a maximal chain of leftmodular elements (Hugh Thomas).
Symmetric Functions and Cylindric Schur Functions
Peter McNamara 3
Outline
Symmetric functions
Schur functions and LittlewoodRichardson 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
:LittlewoodRichardson 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 takehome fact!)
Consequences:
We say that s
µ
s
ν
=
λ
c
λ
µν
s
λ
is a
Schurpositive
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 LittlewoodRichardson rule
LittlewoodRichardson 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 LittlewoodRichardson rule
LittlewoodRichardson 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 LittlewoodRichardson rule
LittlewoodRichardson 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 Schurpositive.
The coefcients in the expansion are the LittlewoodRichar dson
coefcients c
λ
µν
.
c
λ
µν
= number of points of Gr
kn
in
˜
Ω
µ
∩
˜
Ω
ν
∩
˜
Ω
λ
∨
.
The LittlewoodRichardson 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 Schurpositivity research project
Know s
µ
s
ν
=
λ
c
λ
µν
s
λ
is Schurpositive.
Question
Given µ,ν,when is
s
σ
s
τ
−s
µ
s
ν
Schurpositive?In other words,when is c
λ
στ
−c
λ
µν
≥ 0 for
every
partition λ.
Fomin,Fulton,Li,Poon:Eigenvalues,singular values,and
LittlewoodRichardson 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 Schurpositivity research project
Know s
µ
s
ν
=
λ
c
λ
µν
s
λ
is Schurpositive.
Question
Given µ,ν,when is
s
σ
s
τ
−s
µ
s
ν
Schurpositive?In other words,when is c
λ
στ
−c
λ
µν
≥ 0 for
every
partition λ.
Fomin,Fulton,Li,Poon:Eigenvalues,singular values,and
LittlewoodRichardson 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
logconcavity 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,CiocanFontanine,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 GromovWitten 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
= 3point
GromovWitten 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 GromovWitten 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
= 3point
GromovWitten 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 Schurpositive.
Known:
s
µ/d/ν
(x
1
,x
2
,...) ≡ s
µ/d/ν
(x) need not be Schurpositive.
Example
If s
µ/d/ν
= s
22
+s
211
−s
1111
,then s
µ/d/ν
(x
1
,x
2
,x
3
) is Schurpositive.
(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
Schurpositive?
k
n−k
Theorem (McN.)
For any cylindric skew shape C,
s
C
(x
1
,x
2
,...) is Schurpositive ⇔C is a skew shape.
Equivalently,if C is a nontrivial cylindric skew shape,then
s
C
(x
1
,x
2
,...) is
not
Schurpositive.
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,CiocanFontanine,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,CiocanFontanine,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,CiocanFontanine,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,CiocanFontanine,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,CiocanFontanine,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,CiocanFontanine,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,CiocanFontanine,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 Schurpositive 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 Schurpositive 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 Schurpositive
.
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 Schurpositive
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 Schurpositive
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
Schurpositive.
Symmetric Functions and Cylindric Schur Functions
Peter McNamara 25
Summary of results
Classication of those cylindric skew Schur functions that are
Schurpositive.
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 Schurpositive.
Outlook
Prove the conjecture.
Develop a LittlewoodRichardson 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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