Crystal approach to a ne Schubert calculus · Crystal on a ne factorizations + Young (Specht) modules A ne Stanley symmetric functions vs dual k-Schur functions Gromov-Wittens for

Crystal approach to affine Schubert calculus

by Jennifer Morse♣ and Anne Schilling♠

♣ Drexel University ♠ UC Davis

IMRN 2015, doi:10.1093/imrn/rnv194IMRN

Raleigh, North CarolinaOctober 11, 2015

(Raleigh 2015) October 11, 2015 1 / 28

Outline

�� Littlewood-Richardson numbers + variations�� Crystal on affine factorizations + Young (Specht) modules�� Affine Stanley symmetric functions vs dual k-Schur functions�� Gromov-Wittens for flags, Schur times Schubert, etc

(Raleigh 2015) October 11, 2015 2 / 28

Outline

(Raleigh 2015) October 11, 2015 2 / 28

Littlewood-Richardson coefficients cνλµ

Indexed by partitions:

Tensor product multiplicities

V (λ)⊗ V (µ) =⊕ν

cνλµ V (ν)

Symmetric function coefficients

sν/µ =∑λ

cνλµ sλ

Intersections in the Grassmannian

cνλµ = Xλ ∩ Xµ ∩ Xν∨

Structure constants for cohomology of the Grassmannian

σλ ∪ σµ =∑ν⊂rect

cνλµ σν(Raleigh 2015) October 11, 2015 3 / 28

Combinatorial description

Littlewood–Richardson rulecνλµ = # skew tableaux t of shape ν/λ and weight µ such that row(t) is areverse lattice word.

Example

s s = · · ·+?s + · · ·

21

1211

12

1121

11

2112 ⇒ c32121,21 = 2

Gordon James (1987) on the Littlewood-Richardson rule:

“Unfortunately the Littlewood-Richardson rule is much harder toprove than was at first suspected. The author was once told thatthe Littlewood-Richardson rule helped to get men on the moonbut was not proved until after they got there.”

(Raleigh 2015) October 11, 2015 4 / 28

Example

s s = · · ·+?s + · · ·

21

1211

12

1121

11

2112 ⇒ c32121,21 = 2

(Raleigh 2015) October 11, 2015 4 / 28

Example

s s = · · ·+?s + · · ·

21

1211

12

1121

11

2112 ⇒ c32121,21 = 2

(Raleigh 2015) October 11, 2015 4 / 28

Example

s s = · · ·+?s + · · ·

21

1211

12

1121

11

2112 ⇒ c32121,21 = 2

(Raleigh 2015) October 11, 2015 4 / 28

Example

s s = · · ·+?s + · · ·

21

1211

12

1121

11

2112 ⇒ c32121,21 = 2

(Raleigh 2015) October 11, 2015 4 / 28

Crystal graph

Action of crystal operators ei , fi , si on tableaux:

1 Consider letters i and i + 1 in row reading word of the tableau

2 Successively “bracket” pairs of the form (i + 1, i)

3 Left with word of the form i r (i + 1)s

ei (ir (i + 1)s) =

{i r+1(i + 1)s−1 if s > 0

0 else

fi (ir (i + 1)s) =

{i r−1(i + 1)s+1 if r > 0

0 else

si (ir (i + 1)s) = i s(i + 1)r

(Raleigh 2015) October 11, 2015 5 / 28

Crystal graph

Action of crystal operators ei , fi , si on tableaux:

1 Consider letters i and i + 1 in row reading word of the tableau

2 Successively “bracket” pairs of the form (i + 1, i)

3 Left with word of the form i r (i + 1)s

ei (ir (i + 1)s) =

{i r+1(i + 1)s−1 if s > 0

0 else

fi (ir (i + 1)s) =

{i r−1(i + 1)s+1 if r > 0

0 else

si (ir (i + 1)s) = i s(i + 1)r

(Raleigh 2015) October 11, 2015 5 / 28

Crystal reformulation

31 2 2 3

1 1 2 3 3 3

e2: change leftmost unpaired 3 into 2f2: change rightmost unpaired 2 into 3

Theorem

b where all ei (b) = 0 (highest weight)↔ connected component↔ irreducible

(Raleigh 2015) October 11, 2015 6 / 28

31 2 2 3

1 1 2 3 3 3

Theorem

(Raleigh 2015) October 11, 2015 6 / 28

31 2 2 3

1 1 2 3 3 3

→ e2 →← f2 ←

31 2 2 3

1 1 2 2 3 3

Theorem

(Raleigh 2015) October 11, 2015 6 / 28

31 2 2 3

1 1 2 3 3 3

→ e2 →← f2 ←

31 2 2 3

1 1 2 2 3 3

Theorem

(Raleigh 2015) October 11, 2015 6 / 28

31 2 2 3

1 1 2 3 3 3

→ e2 →← f2 ←

31 2 2 3

1 1 2 2 3 3

Theorem

Reformulation of LR rule

cνλµ counts tableaux of shape ν/λ and weight µ which are highest weight.

(Raleigh 2015) October 11, 2015 6 / 28

31 2 2 3

1 1 2 3 3 3

→ e2 →← f2 ←

31 2 2 3

1 1 2 2 3 3

Theorem

Mechanism to get Schur expansion

sν/λ =∑

T∈B(ν/λ)

xweight(T ) =∑

YT=highest weights

sweight(YT )

(Raleigh 2015) October 11, 2015 6 / 28

Decomposition

1 ⊗2 33

2 ⊗2 33

1 ⊗2 23

2 ⊗1 33

2 ⊗1 32

2 ⊗1 222 ⊗

1 23

3 ⊗2 33

3 ⊗1 13

3 ⊗1 12

2 ⊗1 13

3 ⊗2 23

1 ⊗1 12

1 ⊗1 13

1 ⊗1 32

1 ⊗1 33

1 ⊗1 23

1 ⊗1 22

2 ⊗2 23

3 ⊗1 23

3 ⊗1 22

2 ⊗1 12

3 ⊗1 32

3 ⊗1 33

2

1

1 1

1

2

1

2

1

21

2

1

2

1

1 1

2

(Raleigh 2015) October 11, 2015 7 / 28

Variation cwuv

Indexed by permutations: (1,2,3) (2,1,3) (3,2,1) · · ·

Intersections in the set Fn of complete flags0 = W0 ⊂W1 ⊂ · · · ⊂Wn = Cn

cwuv = Xu ∩ Xv ∩ Xw0w

Cohomology of the flag variety structure constants

σu ∪ σv =∑w∈Sn

cwuv σw0w

Schubert polynomial coefficients

Su Sv =∑w

cwuv Sw0w

WHAT ARE THESE COUNTING?(Raleigh 2015) October 11, 2015 8 / 28

Variation cwuv

Indexed by permutations: (1,2,3) (2,1,3) (3,2,1) · · ·

Intersections in the set Fn of complete flags0 = W0 ⊂W1 ⊂ · · · ⊂Wn = Cn

cwuv = Xu ∩ Xv ∩ Xw0w

Cohomology of the flag variety structure constants

σu ∪ σv =∑w∈Sn

cwuv σw0w

Schubert polynomial coefficients

Su Sv =∑w

cwuv Sw0w

WHAT ARE THESE COUNTING?(Raleigh 2015) October 11, 2015 8 / 28

Variations quantized

Grassmannian Flags

partitions in a rectangle permutations

Gromov-Witten invariantscount equivalence classes of rational curves of multidegree d

quantum cohomology

σλ∗qσµ =∑ν⊂rect

qd 〈λ, µ, ν〉d σν σu∗qσv =∑w∈Sn

qd 〈u, v ,w〉d σw0w

polynomial coefficients modulo an ideal

Schur functions sλ quantum Schubert polynomials

Λ Z[x1, . . . , xn; q1, . . . , qn−1]

(Raleigh 2015) October 11, 2015 9 / 28

Variations quantized

Grassmannian Flags

partitions in a rectangle permutations

Gromov-Witten invariantscount equivalence classes of rational curves of multidegree d

quantum cohomology

σλ∗qσµ =∑ν⊂rect

qd 〈λ, µ, ν〉d σν σu∗qσv =∑w∈Sn

qd 〈u, v ,w〉d σw0w

polynomial coefficients modulo an ideal

Schur functions sλ quantum Schubert polynomials

Λ Z[x1, . . . , xn; q1, . . . , qn−1]

(Raleigh 2015) October 11, 2015 9 / 28

Crystals and affine Schubert calculus

(Raleigh 2015) October 11, 2015 10 / 28

Stable Schubert polynomials Fw

restriction: S1m×w −→ Stanley symmetric functions Fw for w ∈ Sn

for 321-avoiding w ,

Fw = sν/µ =∑λ

cνλµ sλ

symmetric and Schur positive

Fw =∑λ

awλ sλ

coefficient of x1x2 · · · xr counts reduced words of w

Sn = 〈s1, . . . , sn−1〉 si sj = sjsi si si+1si = si+1si si+1 s2i = id

(3, 2, 1, 4) = s1s2s1 = s2s1s2 = s3s3s1s2s1

(Raleigh 2015) October 11, 2015 11 / 28

Fw = sν/µ =∑λ

cνλµ sλ

Fw =∑λ

awλ sλ

(3, 2, 1, 4) = s1s2s1 = s2s1s2 = s3s3s1s2s1

(Raleigh 2015) October 11, 2015 11 / 28

Fw = sν/µ =∑λ

cνλµ sλ

Fw =∑λ

awλ sλ

(3, 2, 1, 4) = s1s2s1 = s2s1s2 = s3s3s1s2s1

(Raleigh 2015) October 11, 2015 11 / 28

Fw = sν/µ =∑λ

cνλµ sλ

Fw =∑λ

awλ sλ

(3, 2, 1, 4) = s1s2s1 = s2s1s2 = s3s3s1s2s1

(Raleigh 2015) October 11, 2015 11 / 28

Stable Schubert polynomials

Fw =∑

v r ···v1=w

x`(v1)1 · · · x `(v

r )r

Decreasing factorization of w

1 w is the product of permutations v r · · · v1

2 each v i has a strictly decreasing reduced word

3 `(w) = `(v r ) + · · ·+ `(v1)

w = (2, 1, 4, 3) = s1s3 = s3s1:

(s1)(s3) −→ x1x2(s3)(s1) −→ x1x2()(s3s1) −→ x21(s3s1)() −→ x22

F(2,1,4,3) = 2 x1x2 + x21 + x

22

(Raleigh 2015) October 11, 2015 12 / 28

Stable Schubert polynomials

Fw =∑

v r ···v1=w

x`(v1)1 · · · x `(v

r )r

Decreasing factorization of w

1 w is the product of permutations v r · · · v1

2 each v i has a strictly decreasing reduced word

3 `(w) = `(v r ) + · · ·+ `(v1)

w = (2, 1, 4, 3) = s1s3 = s3s1:

(s1)(s3) −→ x1x2(s3)(s1) −→ x1x2()(s3s1) −→ x21(s3s1)() −→ x22

F(2,1,4,3) = 2 x1x2 + x21 + x

22

(Raleigh 2015) October 11, 2015 12 / 28

Affine Stanley symmetric functions

indexed by affine permutations

Affine symmetric group

〈s0, s1, . . . , sn−1〉 with s0 and sn−1 adjacent

for n = 3, s1s2s1s0 = s2s1s2s0 (s2s0 6= s0s2)s2s0s2 = s0s2s0

(Raleigh 2015) October 11, 2015 13 / 28

indexed by affine permutations introduced by Lam

F̃w =∑

v r ···v1=w

x`(v1)1 · · · x `(v

r )r

Affine factorizations of w

w is a product of affine permutations v r · · · v1

each v i has a reduced word with no j − 1 preceeding jand no n − 1 preceeding 0

`(w) = `(v1) + · · ·+ `(v r )

some affine factorizations of w = s3s2s3s1s0 ∈ S̃4(s3)(s2)(s3)(s1s0) −→ x21x2x3x4(s2)(s3)(s2)(s1s0) −→ x21x2x3x4

(s2)(s3)(s2s1s0) −→ x31x2x3(s2)(s3s2s1s0) is BAD

(Raleigh 2015) October 11, 2015 14 / 28

indexed by affine permutations introduced by Lam

F̃w =∑

v r ···v1=w

x`(v1)1 · · · x `(v

r )r

Affine factorizations of w

w is a product of affine permutations v r · · · v1

each v i has a reduced word with no j − 1 preceeding jand no n − 1 preceeding 0

`(w) = `(v1) + · · ·+ `(v r )

some affine factorizations of w = s3s2s3s1s0 ∈ S̃4(s3)(s2)(s3)(s1s0) −→ x21x2x3x4(s2)(s3)(s2)(s1s0) −→ x21x2x3x4

(s2)(s3)(s2s1s0) −→ x31x2x3(s2)(s3s2s1s0) is BAD

(Raleigh 2015) October 11, 2015 14 / 28

Crystal operators on factorizations

Recall ẽi pairing and action:

31 2 2 3

1 1 2 3 3 3

pairing−→3

1 2 2 31 1 2 3 3 3

ẽ2−→3

1 2 2 31 1 2 2 3 3

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

pairing−→ (9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

from big to small:pair x ∈ 3’s with smallest y ∈ 2’s that is bigger than xdelete smallest unpaired z ∈ 3’s and add z − t to 2’s

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

Label cells diagonally

31 2 2 3

1 1 2 3 3 3

pairing−→30

12 23 24 3514 15 26 37 38 39

ẽ2−→30

12 23 24 3514 15 26 27 38 39

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

31 2 2 3

1 1 2 3 3 3

pairing−→30

12 23 24 3514 15 26 37 38 39

ẽ2−→30

12 23 24 3514 15 26 27 38 39

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

31 2 2 3

1 1 2 3 3 3

pairing−→30

12 23 24 3514 15 26 37 38 39

ẽ2−→30

12 23 24 3514 15 26 27 38 39

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

from big to small:pair x ∈ 3’s with smallest y ∈ 2’s that is bigger than x

delete smallest unpaired z ∈ 3’s and add z − t to 2’s

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

31 2 2 3

1 1 2 3 3 3

pairing−→30

12 23 24 3514 15 26 37 38 39

ẽ2−→30

12 23 24 3514 15 26 27 38 39

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

31 2 2 3

1 1 2 3 3 3

pairing−→30

12 23 24 3514 15 26 37 38 39

ẽ2−→30

12 23 24 3514 15 26 27 38 39

(9 8 7 5 0)︸︷︷︸label of 3’s

(6 4 3)︸︷︷︸label of 2’s

ẽ2−→ (9 8 5 0)︸︷︷︸label of 3’s

(7 6 4 3)︸︷︷︸label of 2’s

operator ẽi

(9 8 7 5 4 3 0)(8 5 4 10)→ (9 8 7 4 3 0)(8 5 4 3 10)

(Raleigh 2015) October 11, 2015 15 / 28

Crystal Theorem

Definition

Fix w ∈ 〈s0, . . . , ŝx , . . . , sn−1〉 = Sx̂ .Graph B(w)

1 vertices are affine factorizations of w

2 edges are imposed and colored by f̃i , ẽi3 highest weights are vertices with no unpaired entries

Theorem (with Morse)

B(w) is a crystal graph of type A`

Proof

Checking Stembridge local axioms

(Raleigh 2015) October 11, 2015 16 / 28

Crystal Theorem

Definition

Proof

(Raleigh 2015) October 11, 2015 16 / 28

Crystal Theorem

Definition

Proof

(Raleigh 2015) October 11, 2015 16 / 28

Examples

[s1, 1, s3][1, s3, s1]

[s3s1, 1, 1]

[1, 1, s3s1]

[s1, s3, 1][1, s3s1, 1]

[1, s1, s3]

[s3, s1, 1]

[s3, 1, s1]

2

21

1

2

12

[1, 1, s2s1]

[1, s2s1, 1]

[1, s2, s1]

[s2, s1, 1]

[s2, 1, s1]

[s2s1, 1, 1]

2

1

1 2

12

(Raleigh 2015) October 11, 2015 17 / 28

Schur expansion

Fix w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉 ⊂ S̃n

F̃w =∑λ

awλ sλ

awλ counts highest weights vr · · · v1 of B(w) with (`(v1), . . . , `(v r )) = λ

In S̃4 (where 0 > 3):[1, s0s2] [s2, s0]

[s2s0, 1]

[s0, s2]

1

=⇒ F̃s2s0 = s2 + s1,1

(Raleigh 2015) October 11, 2015 18 / 28

Schur expansion

Fix w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉 ⊂ S̃n

F̃w =∑λ

awλ sλ

awλ counts highest weights vr · · · v1 of B(w) with (`(v1), . . . , `(v r )) = λ

In S̃4 (where 0 > 3):[1, s0s2] [s2, s0]

[s2s0, 1]

[s0, s2]

1

=⇒ F̃s2s0 = s2 + s1,1

(Raleigh 2015) October 11, 2015 18 / 28

Generalized Young (Specht) modules

Dw diagram of permutation

Theorem (Kraśkiewicz, Reiner-Shimozono; 1995)

awλ = the multiplicity of the irreducible Sn-representation Sλ in the

generalized Young (Specht) module MDw , for w ∈ Sn and λ ` `(w).

crystal interpretation for non-skew shapes!

For any permutation w̃ ∈ Sx̂ ⊂ S̃n, the crystal isomorphism

B(w̃) ∼=⊕λ

B(λ)⊕awλ

is explicitly given by the Edelman-Greene insertion ϕQEG(v` · · · v1) = Q:

ϕQEG ◦ ẽi = ẽi ◦ ϕQEG

(Raleigh 2015) October 11, 2015 19 / 28

Generalized Young (Specht) modules

Dw diagram of permutation

Theorem (Kraśkiewicz, Reiner-Shimozono; 1995)

awλ = the multiplicity of the irreducible Sn-representation Sλ in the

generalized Young (Specht) module MDw , for w ∈ Sn and λ ` `(w).

crystal interpretation for non-skew shapes!

For any permutation w̃ ∈ Sx̂ ⊂ S̃n, the crystal isomorphism

B(w̃) ∼=⊕λ

B(λ)⊕awλ

is explicitly given by the Edelman-Greene insertion ϕQEG(v` · · · v1) = Q:

ϕQEG ◦ ẽi = ẽi ◦ ϕQEG

(Raleigh 2015) October 11, 2015 19 / 28

Example

Crystal of type A2 for w0 = s1s2s1 ∈ S3

(s1, 1, s2s1)

(s1, s2s1, 1)(s2s1, 1, s2)

(s2s1, s2, 1)

(1, s2s1, s2)

(s1, s2, s1)(s2, s1, s2)

(1, s1, s2s1)

1

2

1

2

1

(Raleigh 2015) October 11, 2015 20 / 28

(Raleigh 2015) October 11, 2015 21 / 28

Dual k-Schur functions Fkν/λ

symmetric functions indexed by skew shapes ν/λ (for k-bounded ν, λ)

when k is big, Fkν/λ = sν/λ

straight shape elements {Fkλ} form basis for quotient in Λ

{Fkλ} are representatives for cohomology classes of the affineGrassmannian Gr = SL(n,C((t)))/SL(n,C[[t]]) (Lam)

expansion coefficients of Fkν/λ in terms of straight case elements areGromov-Witten invariants (Lapointe, Morse)

(Raleigh 2015) October 11, 2015 22 / 28

Applications of straight shape expansion

Fkν/λ =∑µ

Cλµν Fkµ � σλ ∗q σµ =∑ν⊂rect

|ν|=|λ|+|µ|−dn

qd (WZW fusion)σν

↓

σu ∗q σv =∑w

∑d

qd (flag GW) σw0w

�

�Computation in Λ

Coefficients Cλµν when hook(µ) < n include

k-Schur coefficients in product of Schur times k-Schur

all fusion coefficients

coefficients in Schur times a Schubert polynomial

Gromov-Witten invariants for flags 〈u, v ,w〉d where u has one descentSchubert decomposition of positroid varieties

(Raleigh 2015) October 11, 2015 23 / 28

Applications of straight shape expansion

Fkν/λ =∑µ

Cλµν Fkµ � σλ ∗q σµ =∑ν⊂rect

|ν|=|λ|+|µ|−dn

qd (WZW fusion)σν

↓

σu ∗q σv =∑w

∑d

qd (flag GW) σw0w

�

�Computation in Λ

Coefficients Cλµν when hook(µ) < n include

k-Schur coefficients in product of Schur times k-Schur

all fusion coefficients

coefficients in Schur times a Schubert polynomial

Gromov-Witten invariants for flags 〈u, v ,w〉d where u has one descentSchubert decomposition of positroid varieties

(Raleigh 2015) October 11, 2015 23 / 28

Affine Stanley/dual k-Schur correspondence

LC : {wλ affine Grassmannian permutation} ↔ {λ k-bounded}elements where every word ends in s0

LC : wλ 7→ λ = linv(wλ)′

Ex: . . . 2 -7 -5 -4 -1 7 [-2,0,1,4,12] 7→ (3, 2, 2, 1, 0)′ = (4, 3, 1)

Grassmannian case of affine Stanley F̃wλ = straight shape dual k-Schur Fkλ

bijection κ: {w generic affine element } ↔ {skew k-bounded ν/λ}generic affine Stanley F̃w = dual k-Schur Fkν/λ

κ involves decomposition w̃ = ṽu for w̃ ∈ S̃n, ṽ ∈ S̃0n , u ∈ Snaf : Sn → S̃n,(n2)

u 7→ [u(1), u(2) + n, . . . , u(n) + (n − 1)n]

(Raleigh 2015) October 11, 2015 24 / 28

Ex: . . . 2 -7 -5 -4 -1 7 [-2,0,1,4,12] 7→ (3, 2, 2, 1, 0)′ = (4, 3, 1)

u 7→ [u(1), u(2) + n, . . . , u(n) + (n − 1)n]

(Raleigh 2015) October 11, 2015 24 / 28

Ex: . . . 2 -7 -5 -4 -1 7 [-2,0,1,4,12] 7→ (3, 2, 2, 1, 0)′ = (4, 3, 1)

u 7→ [u(1), u(2) + n, . . . , u(n) + (n − 1)n]

(Raleigh 2015) October 11, 2015 24 / 28

Ex: . . . 2 -7 -5 -4 -1 7 [-2,0,1,4,12] 7→ (3, 2, 2, 1, 0)′ = (4, 3, 1)

u 7→ [u(1), u(2) + n, . . . , u(n) + (n − 1)n]

(Raleigh 2015) October 11, 2015 24 / 28

Consider

Fkν/λ =∑µ

CλµνFkµ =∑µ

Cλµνsµ if hook(µ) < n

=F̃w =∑µ

awµsµ

For any w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉Cλµν = # of affine factorizations of w with weight µ killed by all ẽi .

(Raleigh 2015) October 11, 2015 25 / 28

Consider

Fkν/λ =∑µ

CλµνFkµ =∑µ

Cλµνsµ if hook(µ) < n

=F̃w =∑µ

awµsµ

For any w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉Cλµν = # of affine factorizations of w with weight µ killed by all ẽi .

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Enumerated by highest weight affine factorizations:

k-Schur expansion of sµs(k)w

coefficients of s(k)v when hook(µ) < n and wv−1 ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉

Schubert polynomial expansion of sµSw

for any w ∈ Sn and partition µ where |µc | < n

Fusion rules Nνλµ

ν/λ has a cut-point OR

µ satisfies |µc | < n.

Gromov-Witten invariants 〈u,w , v〉d for complete flagswhen u has one descent at position r and afd(v)af(w)

−1 ∈ Sx̂

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−1 ∈ Sx̂

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−1 ∈ Sx̂

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−1 ∈ Sx̂

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Future Work

Gromov-Witten invariantsCloser study of crystal structure on affine factorizations and crystaloperators on dual k-tableaux

t-analogue of k-Schur functions and relation to energy on KR crystals(charge plus offset) ⇒ generalization to other typesOther types

K -theory analogue of the crystal operators

Thank you !

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Future Work

Thank you !

(Raleigh 2015) October 11, 2015 28 / 28

Future Work

Thank you !

(Raleigh 2015) October 11, 2015 28 / 28

Future Work

Thank you !

(Raleigh 2015) October 11, 2015 28 / 28

Future Work

Thank you !

(Raleigh 2015) October 11, 2015 28 / 28

Crystal approach to a ne Schubert calculus · Crystal on a ne factorizations + Young (Specht) modules A ne Stanley symmetric functions vs dual k-Schur functions Gromov-Wittens for

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