Post on 25-Aug-2020
Catalan functions and k-Schur positivity
Jonah Blasiak
Drexel University
joint work with Jennifer Morse, Anna Pun, and Dan Summers
April 2018
Strengthened Macdonald positivity conjecture
Theorem (Haiman)
The modified Macdonald polynomials are Schur positive:
Hµ(x; q, t) =∑λ
Kλµ(q, t)sλ(x) for Kλµ(q, t) ∈ N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The atom k-Schur functions {Aλ(x; t)}λ1≤k• form a basis for Λk = spanQ(q,t){Hµ(x; q, t)}µ1≤k , and
• are Schur positive;
• expansion of Hµ(x; q, t) ∈ Λk in this basis has coefficients in N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The k + 1-Schur expansion of a k-Schur function has coefficients in N[t] .
Strengthened Macdonald positivity conjecture
Theorem (Haiman)
The modified Macdonald polynomials are Schur positive:
Hµ(x; q, t) =∑λ
Kλµ(q, t)sλ(x) for Kλµ(q, t) ∈ N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The atom k-Schur functions {Aλ(x; t)}λ1≤k• form a basis for Λk = spanQ(q,t){Hµ(x; q, t)}µ1≤k , and
• are Schur positive;
• expansion of Hµ(x; q, t) ∈ Λk in this basis has coefficients in N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The k + 1-Schur expansion of a k-Schur function has coefficients in N[t] .
Strengthened Macdonald positivity conjecture
Theorem (Haiman)
The modified Macdonald polynomials are Schur positive:
Hµ(x; q, t) =∑λ
Kλµ(q, t)sλ(x) for Kλµ(q, t) ∈ N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The atom k-Schur functions {Aλ(x; t)}λ1≤k• form a basis for Λk = spanQ(q,t){Hµ(x; q, t)}µ1≤k , and
• are Schur positive;
• expansion of Hµ(x; q, t) ∈ Λk in this basis has coefficients in N[q, t].
Conjecture (Lapointe-Lascoux-Morse)
The k + 1-Schur expansion of a k-Schur function has coefficients in N[t] .
Strengthened Macdonald positivity conjecture
Example. k = 2
H14 = t4(s + ts + t2s
)+(t2 + t3
)(s + ts
)+
(s + ts + t2s
)H211 = t
(s + ts + t2s
)+(1 + qt2
)(s + ts
)+ q
(s + ts + t2s
)H22 =
(s + ts + t2s
)+ (tq + q)︸ ︷︷ ︸
positive sum ofq, t-monomials
(s + ts
)︸ ︷︷ ︸
t-positive sumof schur functions
+q2
(s + ts + t2s
)
︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸s
(2)s
(2)s
(2)
basis for restricted span Λk of Macdonald polynomials
Strengthened Macdonald positivity conjecture
Example. k = 2
H14 = t4(s + ts + t2s
)+(t2 + t3
)(s + ts
)+
(s + ts + t2s
)H211 = t
(s + ts + t2s
)+(1 + qt2
)(s + ts
)+ q
(s + ts + t2s
)H22 =
(s + ts + t2s
)+ (tq + q)︸ ︷︷ ︸
positive sum ofq, t-monomials
(s + ts
)︸ ︷︷ ︸
t-positive sumof schur functions
+q2
(s + ts + t2s
)
︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸s
(2)s
(2)s
(2)
basis for restricted span Λk of Macdonald polynomials
Conjecturally equivalent definitions of k-Schurs
Schurbasis symmetric positive branching
[1998:Lapointe,Lascoux,Morse] X XTableaux and katabolism[2003:Lapointe,Morse] X XJing vertex operators[2006:Lam,Lapointe,Morse,Shimozono]
Bruhat order on type-Aaffine Weyl group / strong tableaux[2010:Chen,Haiman] XGL`(C)-equivariant Eulercharacteristics / Demazure operators[2012:Assaf,Billey]
Quasisymmetric functions[2015:Dalal,Morse] X XInverting affine Kostka matrix
Overview
The k-Schur functions appear in
• the study of Macdonald polynomials,
• the homology of the affine Grassmannian,
• graded representations of the symmetric group.
Prior work on the branching rule:
• Geometric proof at t = 1 (Lam 2011).
• Formula for branching at t = 1 as equivalence classes on the k-shapeposet (Lam-Lapointe-Morse-Shimozono 2013).
Main results:
• Strong tableaux k-Schur functions form a Schur positive basis for Λk .
• (Branching rule) positive combinatorial formula for the k + 1-Schurexpansion of k-Schur functions.
• Strong tableaux k-Schur functions agree with a Catalan functiondefinition of Chen-Haiman.
Overview
The k-Schur functions appear in
• the study of Macdonald polynomials,
• the homology of the affine Grassmannian,
• graded representations of the symmetric group.
Prior work on the branching rule:
• Geometric proof at t = 1 (Lam 2011).
• Formula for branching at t = 1 as equivalence classes on the k-shapeposet (Lam-Lapointe-Morse-Shimozono 2013).
Main results:
• Strong tableaux k-Schur functions form a Schur positive basis for Λk .
• (Branching rule) positive combinatorial formula for the k + 1-Schurexpansion of k-Schur functions.
• Strong tableaux k-Schur functions agree with a Catalan functiondefinition of Chen-Haiman.
Overview
The k-Schur functions appear in
• the study of Macdonald polynomials,
• the homology of the affine Grassmannian,
• graded representations of the symmetric group.
Prior work on the branching rule:
• Geometric proof at t = 1 (Lam 2011).
• Formula for branching at t = 1 as equivalence classes on the k-shapeposet (Lam-Lapointe-Morse-Shimozono 2013).
Main results:
• Strong tableaux k-Schur functions form a Schur positive basis for Λk .
• (Branching rule) positive combinatorial formula for the k + 1-Schurexpansion of k-Schur functions.
• Strong tableaux k-Schur functions agree with a Catalan functiondefinition of Chen-Haiman.
Conjecturally equivalent definitions of k-Schurs
Schurbasis symmetric positive branching
[1998:Lapointe,Lascoux,Morse] X XTableaux and katabolism[2003:Lapointe,Morse] X XJing vertex operators[2006:Lam,Lapointe,Morse,Shimozono]
Bruhat order on type-Aaffine Weyl group / strong tableaux[2010:Chen,Haiman] XCatalan functions[2012:Assaf,Billey]
Quasisymmetric functions[2015:Dalal,Morse] X XInverting affine Kostka matrix[2018:B,Morse,Pun,Summers] X X X XStrong tableaux = Catalan functions
k-bounded partitions and k + 1-cores
Def. A k-bounded partition is a partition with parts of size ≤ k .
Def. A k + 1-core is a partition whose diagram has no box with hooklength k + 1.
Proposition. There is a bijection κ 7→ p(κ) from k + 1-cores tok-bounded partitions.
Example. k = 4.14 12 9 7 6 4 3 2 1
9 7 4 2 1
6 4 1
4 2
3 1
1
κ p(κ)
Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained byremoving boxes of hook length > k .
k-bounded partitions and k + 1-cores
Def. A k-bounded partition is a partition with parts of size ≤ k .
Def. A k + 1-core is a partition whose diagram has no box with hooklength k + 1.
Proposition. There is a bijection κ 7→ p(κ) from k + 1-cores tok-bounded partitions.
Example. k = 4.14 12 9 7 6 4 3 2 1
9 7 4 2 1
6 4 1
4 2
3 1
1
κ p(κ)
Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained byremoving boxes of hook length > k .
k-bounded partitions and k + 1-cores
Def. A k-bounded partition is a partition with parts of size ≤ k .
Def. A k + 1-core is a partition whose diagram has no box with hooklength k + 1.
Proposition. There is a bijection κ 7→ p(κ) from k + 1-cores tok-bounded partitions.
Example. k = 4.4 3 2 1
4 2 1
4 1
4 2
3 1
1
k-skew(κ) p(κ)
Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained byremoving boxes of hook length > k .
Strong covers
Def. An inclusion τ ⊂ κ of k + 1-cores is a strong cover, denoted τ ⇒ κ,if |p(τ)|+ 1 = |p(κ)|.
Example.Strong cover with k = 4: corresponding k-skew diagrams:
=⇒
• •••
• •••
=⇒
p(τ) = 332221111 p(κ) = 222222221
Strong covers
Def. An inclusion τ ⊂ κ of k + 1-cores is a strong cover, denoted τ ⇒ κ,if |p(τ)|+ 1 = |p(κ)|.
Example.Strong cover with k = 4: corresponding k-skew diagrams:
=⇒
• •••
• •••
=⇒
p(τ) = 332221111 p(κ) = 222222221
Strong marked covers
Def. A strong marked cover τr
==⇒ κ is a strong cover τ ⇒ κ togetherwith a positive integer r which is allowed to be the smallest row index ofany connected component of the skew shape κ/τ .
Example. The two possible markings of the previous strong cover:
• •••
• •?••
• •?••
• •••
τ6
==⇒ κ τ3
==⇒ κ
Spin
Def.spin
(τ
r==⇒ κ
)= c · (h − 1) + N, where
• c = number of connected components of κ/τ ,
• h = height (number of rows) of each component,
• N = number of components below the marked one.
Example.
• •••
• •?••
• •?••
• •••
τ6
==⇒ κ τ3
==⇒ κ
spin = 4 spin = 5
spin = c · (h − 1) + N = 2 · (3− 1) + 0 = 4 spin = 2 · (3− 1) + 1 = 5
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
5
5
5?
κ(4) 5==⇒ κ(5)
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
4
4?
κ(3) 4==⇒ κ(4)
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
3
3?
3
κ(2) 3==⇒ κ(3)
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
2 2 2?
2 κ(1) 2==⇒ κ(2)
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
1?
κ(0) 1==⇒ κ(1)
Vertical strong marked tableaux
Def. A vertical strong marked tableau T of weight η = (η1, η2, . . . ) is asequence
κ(0) r1==⇒ κ(1) r2==⇒ · · · rm===⇒ κ(m)
such that rvi+1 < rvi+2 < · · · < rvi+ηi for all i , where vi := η1 + · · ·+ ηi−1.
• inside(T ) := p(κ(0))
• outside(T ) := p(κ(m))
Example. For k = 4, a vertical strong marked tableau of weight (5):
1? 3 5
2 2 2? 4
2 3? 5
4?
3 5?
Spin k-Schur functions
• We work in the ring of symmetric functions in infinitely manyvariables x = (x1, x2, . . . ).
• SMTkη(µ) = set of strong marked tableaux T of weight η with
outside(T ) = µ.
• spin(T ) = sum of the spins of the strong marked covers comprising T .
Def. For a k-bounded partition µ, let
s(k)µ (x; t) =
∑η∈Z∞≥0, |η|=|µ|
∑T∈SMTk
η(µ)
tspin(T )xη.
Their t = 1 specializations
• agree with another combinatorial definition using weak tableaux(Lam-Lapointe-Morse-Shimozono 2010),
• are Schubert classes in the homology of the affine GrassmannianGrSLk+1
of SLk+1 (Lam 2008).
Spin k-Schur functions
• We work in the ring of symmetric functions in infinitely manyvariables x = (x1, x2, . . . ).
• SMTkη(µ) = set of strong marked tableaux T of weight η with
outside(T ) = µ.
• spin(T ) = sum of the spins of the strong marked covers comprising T .
Def. For a k-bounded partition µ, let
s(k)µ (x; t) =
∑η∈Z∞≥0, |η|=|µ|
∑T∈SMTk
η(µ)
tspin(T )xη.
Their t = 1 specializations
• agree with another combinatorial definition using weak tableaux(Lam-Lapointe-Morse-Shimozono 2010),
• are Schubert classes in the homology of the affine GrassmannianGrSLk+1
of SLk+1 (Lam 2008).
Properties of k-Schur functions
Theorem (B.-Morse-Pun-Summers)
The k-Schur functions {s(k)µ | µ is k-bounded of length ≤ `} satisfy
(vertical dual Pieri rule) e⊥d s(k)µ =
∑T∈VSMTk
(d)(µ)
tspin(T )s(k)inside(T ) ,
(shift invariance) s(k)µ = e⊥` s
(k+1)
µ+1`,
(Schur function stability) if k ≥ |µ|, then s(k)µ = sµ.
• e⊥d ∈ End(Λ) is defined by 〈e⊥d (g), h〉 = 〈g , edh〉 for all g , h ∈ Λ.
• VSMTkη(µ) = set of vertical strong marked tableaux T of weight η
with outside(T ) = µ.
k-Schur branching rule
Theorem (B.-Morse-Pun-Summers)
For µ a k-bounded partition of length ≤ `, the expansion of the k-Schur
function s(k)µ into k + 1-Schur functions is given by
s(k)µ =
∑T∈VSMTk+1
(`)(µ+1`)
tspin(T )s(k+1)inside(T ).
Proof.
The shift invariance property followed by the vertical dual Pieri rule yields
s(k)µ = e⊥` s
(k+1)
µ+1`=
∑T∈VSMTk+1
(`)(µ+1`)
tspin(T )s(k+1)inside(T ).
k-Schur branching rule
Theorem (B.-Morse-Pun-Summers)
For µ a k-bounded partition of length ≤ `, the expansion of the k-Schur
function s(k)µ into k + 1-Schur functions is given by
s(k)µ =
∑T∈VSMTk+1
(`)(µ+1`)
tspin(T )s(k+1)inside(T ).
Proof.
The shift invariance property followed by the vertical dual Pieri rule yields
s(k)µ = e⊥` s
(k+1)
µ+1`=
∑T∈VSMTk+1
(`)(µ+1`)
tspin(T )s(k+1)inside(T ).
k-Schur branching rule
s(3)22221 = t2s
(4)3222 + t2s
(4)3321 + t2s
(4)33111 + s
(4)22221
1? 3 5
2 2 2? 4
2 3? 5
4?
3 5?
1? 3 5
2? 4
1 3? 5
2 4?
3 5?
1? 3 3 5
2? 4
1 3 3? 5
2 4?
5?
1? 3 3 5
2 2? 4
3 3? 5
4?
5?
VSMT4(5)(33332)
k-Schur branching rule
s(3)22221 = t2s
(4)3222 + t2s
(4)3321 + t2s
(4)33111 + s
(4)22221
1? 3 5
2 2 2? 4
2 3? 5
4?
3 5?
1? 3 5
2? 4
1 3? 5
2 4?
3 5?
1? 3 3 5
2? 4
1 3 3? 5
2 4?
5?
1? 3 3 5
2 2? 4
3 3? 5
4?
5?
VSMT4(5)(33332)
T =
1? 3 5
2 2 2? 4
2 3? 5
4?
3 5?
spin(T ) = 0 + 1 + 1 + 0 + 0 = 2 inside(T ) = 3222 outside(T ) = 33332
Root ideals
• Set of positive roots ∆+ :={
(i , j) | 1 ≤ i < j ≤ `}
.
• Ψ ⊆ ∆+ is an upper order ideal of positive roots.
Example. Ψ = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)}
(1, 3) (1, 4) (1, 5) (1, 6)
(2, 5) (2, 6)
(3, 6)
Catalan functions
Def. (Panyushev, Chen-Haiman)
• Ψ ⊆ ∆+ is an upper order ideal of positive roots,
• γ ∈ Z`.The Catalan function indexed by Ψ and γ:
HΨγ (x; t) :=
∏(i , j)∈Ψ
(1− tRij)−1sγ(x)
where the raising operator Rij acts by Rij(sγ(x)) = sγ+εi−εj (x).
Example. Let µ = (µ1, . . . , µ`) be a partition.
• Empty root set: H∅µ (x; t) = sµ(x).
• Full root set: H∆+
µ (x; t) = Hµ(x; t), the modified Hall-Littlewoodpolynomial.
Catalan functions
Def. (Panyushev, Chen-Haiman)
• Ψ ⊆ ∆+ is an upper order ideal of positive roots,
• γ ∈ Z`.The Catalan function indexed by Ψ and γ:
HΨγ (x; t) :=
∏(i , j)∈Ψ
(1− tRij)−1sγ(x)
where the raising operator Rij acts by Rij(sγ(x)) = sγ+εi−εj (x).
Example. Let µ = (µ1, . . . , µ`) be a partition.
• Empty root set: H∅µ (x; t) = sµ(x).
• Full root set: H∆+
µ (x; t) = Hµ(x; t), the modified Hall-Littlewoodpolynomial.
k-Schur Catalan functions
Def. For µ a k-bounded partition of length ≤ `, define the root ideal
∆k(µ) = {(i , j) ∈ ∆+ | k − µi + i < j},
and the Catalan function
s(k)µ (x; t) := H∆k (µ)
µ =∏̀i=1
∏̀j=k+1−µi+i
(1− tRij
)−1sµ(x) .
“ # nonroots in row i = k − µi ”
Examples of Catalan functions
Example. k = 4, µ = 3321.
∆k(µ) =
1, 3 1, 4
2, 4
s(k)µ (x; t) =
∏(i , j)∈∆k (µ)
(1− tRij)−1sµ(x)
Examples of Catalan functions
Example. k = 4, µ = 3321.
∆k(µ) =
1, 3 1, 4
2, 4
s(k)µ (x; t) =
∏(i , j)∈∆k (µ)
(1− tRij)−1sµ(x)
= (1− tR13)−1(1− tR24)−1(1− tR14)−1s3321(x)
Examples of Catalan functions
Example. k = 4, µ = 3321.
∆k(µ) =
1, 3 1, 4
2, 4
s(k)µ (x; t) =
∏(i , j)∈∆k (µ)
(1− tRij)−1sµ(x)
= (1− tR13)−1(1− tR24)−1(1− tR14)−1s3321(x)
= s3321 + t(s3420 + s4311 + s4320) + t2(s4410 + s5301 + s5310)
+ t3(s63−11 + s5400 + s6300) + t4(s64−10 + s73−10)
Examples of Catalan functions
Example. k = 4, µ = 3321.
∆k(µ) =
1, 3 1, 4
2, 4
s(k)µ (x; t) =
∏(i , j)∈∆k (µ)
(1− tRij)−1sµ(x)
= (1− tR13)−1(1− tR24)−1(1− tR14)−1s3321(x)
= s3321 + t(s3420 + s4311 + s4320) + t2(s4410 + s5301 + s5310)
+ t3(s63−11 + s5400 + s6300) + t4(s64−10 + s73−10)
= s3321 + t(s4320 + s4311) + t2(s4410 + s5310) + t3s5400.
Chen-Haiman conjecture
Theorem (B.-Morse-Pun-Summers)
For any k-bounded partition µ, the k-Schur function s(k)µ (x; t) is the
Catalan function s(k)µ (x; t).
k-Schur into Schur
Theorem (B.-Morse-Pun-Summers)
Let µ be a k-bounded partition of length ≤ ` and set m = max(|µ| − k , 0).
The Schur expansion the k-Schur function s(k)µ is given by
s(k)µ =
∑T∈VSMTk+m
(`m)(µ+m`)
tspin(T )sinside(T ).
Proof.
Applying the shift invariance property m times followed by the verticaldual Pieri rule, we obtain
s(k)µ = (e⊥` )m s
(k+m)
µ+m`=
∑T∈VSMTk+m
(`m)(µ+m`)
tspin(T )sinside(T ).
The Schur function stability property ensures this is the Schur functiondecomposition.
k-Schur into Schur
Theorem (B.-Morse-Pun-Summers)
Let µ be a k-bounded partition of length ≤ ` and set m = max(|µ| − k , 0).
The Schur expansion the k-Schur function s(k)µ is given by
s(k)µ =
∑T∈VSMTk+m
(`m)(µ+m`)
tspin(T )sinside(T ).
Proof.
Applying the shift invariance property m times followed by the verticaldual Pieri rule, we obtain
s(k)µ = (e⊥` )m s
(k+m)
µ+m`=
∑T∈VSMTk+m
(`m)(µ+m`)
tspin(T )sinside(T ).
The Schur function stability property ensures this is the Schur functiondecomposition.
Schur expansion of s(1)111 = H111
1? 2 4 4? 5 6
1 2? 4 4 5? 6
3? 5 6?
1 1? 2 4 4? 5 6
2? 4 4 5? 6
3? 5 6?
1? 3 4? 5 5 5 6
2? 5 5 5? 6
1 3? 6?
1 1? 3 4? 5 5 5 6
2? 5 5 5? 6
3? 6?
t3 s3
t2 s21
t s21
s111
s(1)111 = t3s3 + t2s21 + ts21 + s111
The Schur expansion of the 1-Schur function s(1)111 is obtained by summing
tspin(T )sinside(T ) over the set VSMT3(3,3)(3, 3, 3) of vertical strong marked
tableaux T given above.
Schur function straightening
Schur functions may be defined for any γ ∈ Z`. The Schur functionsγ(x1, x2, . . . , x`) = sγ(x) is straightened as follows:
sγ(x) =
{sgn(γ + ρ)ssort(γ+ρ)−ρ(x) if γ + ρ has distinct nonnegative parts,
0 otherwise,
• sort(β) = weakly decreasing sequence obtained by sorting β,
• sgn(β) = sign of the shortest permutation taking β to sort(β).
Example. ` = 4, γ = 3125.
γ + ρ = (3, 1, 2, 5) + (3, 2, 1, 0) = (6, 3, 3, 5) has a repeated part.
Hence s3125(x) = 0.
Schur function straightening
Schur functions may be defined for any γ ∈ Z`. The Schur functionsγ(x1, x2, . . . , x`) = sγ(x) is straightened as follows:
sγ(x) =
{sgn(γ + ρ)ssort(γ+ρ)−ρ(x) if γ + ρ has distinct nonnegative parts,
0 otherwise,
• sort(β) = weakly decreasing sequence obtained by sorting β,
• sgn(β) = sign of the shortest permutation taking β to sort(β).
Example. ` = 4, γ = 3125.
γ + ρ = (3, 1, 2, 5) + (3, 2, 1, 0) = (6, 3, 3, 5) has a repeated part.
Hence s3125(x) = 0.
Schur function straightening
Schur functions may be defined for any γ ∈ Z`. The Schur functionsγ(x1, x2, . . . , x`) = sγ(x) is straightened as follows:
sγ(x) =
{sgn(γ + ρ)ssort(γ+ρ)−ρ(x) if γ + ρ has distinct nonnegative parts,
0 otherwise,
• sort(β) = weakly decreasing sequence obtained by sorting β,
• sgn(β) = sign of the shortest permutation taking β to sort(β).
Example. ` = 4, γ = 4716.
γ + ρ = (4, 7, 1, 6) + (3, 2, 1, 0) = (7, 9, 2, 6)
sort(γ + ρ) = (9, 7, 6, 2)
sort(γ + ρ)− ρ = (6, 5, 5, 2)
Hence s4716(x) = s6552(x).