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Crystal approach to a ne Schubert calculus · Crystal on a ne factorizations + Young (Specht)...
Transcript of Crystal approach to a ne Schubert calculus · Crystal on a ne factorizations + Young (Specht)...
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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
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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
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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
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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
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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
-
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
-
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
-
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
-
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
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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
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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
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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
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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
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Crystal reformulation
31 2 2 3
1 1 2 3 3 3
→ e2 →← f2 ←
31 2 2 3
1 1 2 2 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
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Crystal reformulation
31 2 2 3
1 1 2 3 3 3
→ e2 →← f2 ←
31 2 2 3
1 1 2 2 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
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Crystal reformulation
31 2 2 3
1 1 2 3 3 3
→ e2 →← f2 ←
31 2 2 3
1 1 2 2 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
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
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Crystal reformulation
31 2 2 3
1 1 2 3 3 3
→ e2 →← f2 ←
31 2 2 3
1 1 2 2 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
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
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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
2
1
2
2
1
1
1
21
2
1
2
2
2
1
1 1
2
2
2
(Raleigh 2015) October 11, 2015 7 / 28
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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
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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
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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
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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
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Crystals and affine Schubert calculus
�� ��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 10 / 28
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Crystals and affine Schubert calculus
�� ��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 10 / 28
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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
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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
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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
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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
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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
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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
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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
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Affine Stanley symmetric functions
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
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Affine Stanley symmetric functions
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
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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
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Crystal operators on factorizations
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
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
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Crystal operators on factorizations
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
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
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Crystal operators on factorizations
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
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 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
-
Crystal operators on factorizations
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
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
-
Crystal operators on factorizations
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
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
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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
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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
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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
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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
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
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Schur expansion
Fix w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉 ⊂ S̃n
Theorem (with Morse)
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
1
=⇒ F̃s2s0 = s2 + s1,1
(Raleigh 2015) October 11, 2015 18 / 28
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Schur expansion
Fix w ∈ 〈s0, . . . , sx̂ , . . . , sn−1〉 ⊂ S̃n
Theorem (with Morse)
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
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!
Theorem (with Morse)
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!
Theorem (with Morse)
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
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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
1
2
1
2
2
2
1
(Raleigh 2015) October 11, 2015 20 / 28
-
Crystals and affine Schubert calculus
�� ��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 21 / 28
-
Crystals and affine Schubert calculus
�� ��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 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
-
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
-
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
-
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
-
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
Theorem (with Morse)
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
-
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
Theorem (with Morse)
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
-
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
Theorem (with Morse)
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
-
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
Theorem (with Morse)
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
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Affine Stanley/dual k-Schur correspondence
Consider
Fkν/λ =∑µ
CλµνFkµ =∑µ
Cλµνsµ if hook(µ) < n
=F̃w =∑µ
awµsµ
Theorem (with Morse)
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
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Affine Stanley/dual k-Schur correspondence
Consider
Fkν/λ =∑µ
CλµνFkµ =∑µ
Cλµνsµ if hook(µ) < n
=F̃w =∑µ
awµsµ
Theorem (with Morse)
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
-
Crystals and affine Schubert calculus
�� ��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 26 / 28
-
Crystals and affine Schubert calculus
�� ��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 26 / 28
-
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
(Raleigh 2015) October 11, 2015 27 / 28
-
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
(Raleigh 2015) October 11, 2015 27 / 28
-
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
(Raleigh 2015) October 11, 2015 27 / 28
-
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
(Raleigh 2015) October 11, 2015 27 / 28
-
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 !
(Raleigh 2015) October 11, 2015 28 / 28
-
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 !
(Raleigh 2015) October 11, 2015 28 / 28
-
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 !
(Raleigh 2015) October 11, 2015 28 / 28
-
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 !
(Raleigh 2015) October 11, 2015 28 / 28
-
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 !
(Raleigh 2015) October 11, 2015 28 / 28