A Demazure crystals construction for Schubert polynomials · A Demazure crystals construction for Schubert polynomials Anne Schilling Department ofMathematics, UCDavis basedonjointworkwith

Crystal: Schur functions Crystal: Demazure characters Crystal: Stanley symmetric functions Crystal: Schubert polynomials

A Demazure crystals construction for Schubert

polynomials

Anne Schilling

Department of Mathematics, UC Davis

based on joint work with Sami Assaf in arXiv:1705.09649 [math.CO]

Equivariant Combinatorics, Montreal, June 20, 2017


Schubert polynomials – history

1973: Bernstein–Gel’fand2 introduced certain polynomialrepresentatives of cohomology classes of Schubert cycles Xw in flagvarieties




1982: Lascoux–Schutzenberger via divided difference operators

Sw (x) = ∂w−1w0(xn−1

1 xn−22 · · · xn−1)

w ∈ Sn, w0 = nn − 1 . . . 21 long permutation in Sn

∂i =1− si

xi − xi+1




1982: Lascoux–Schutzenberger via divided difference operators

Sw (x) = ∂w−1w0(xn−1

1 xn−22 · · · xn−1)

w ∈ Sn, w0 = nn − 1 . . . 21 long permutation in Sn

∂i =1− si

xi − xi+1

1993: Billey–Jockusch–Stanley combinatorial expression in terms ofcompatible sequences



1995: Reiner–Shimozono prove statement ofLascoux–Schutzenberger that Schubert polynomials arepositive sums of Demazure characters uses Edelman–Greene insertion



1995: Reiner–Shimozono prove statement ofLascoux–Schutzenberger that Schubert polynomials arepositive sums of Demazure characters uses Edelman–Greene insertion

2017: Assaf new key tableaux expansion of Demazure characters uses weak Edelman–Greene insertion


Stanley symmetric functions –history

1984: Stanley introduced Stanley symmetric functions Fw toenumerate reduced expressions of permutations




Inverse limits of Schubert polynomials

Fw (x1, x2, . . .) = limm→∞

S1m×w (x1, x2, . . . , xn+m)





Fw (x1, x2, . . .) = limm→∞

S1m×w (x1, x2, . . . , xn+m)

Combinatorial formula in terms of decreasing factorizations of w





Fw (x1, x2, . . .) = limm→∞

S1m×w (x1, x2, . . . , xn+m)


1987: Edelman–Greene proved positive Schur expansion of Fw usingEdelman–Greene insertion





Fw (x1, x2, . . .) = limm→∞

S1m×w (x1, x2, . . . , xn+m)


1987: Edelman–Greene proved positive Schur expansion of Fw usingEdelman–Greene insertion

2016: Morse–Schilling imposed crystal structure on decreasingfactorization


Goal

Prove:

Schubert polynomials are Demazure truncations of Stanley symmetricfunctions.


Goal

Prove:


Strategy:

Impose Demazure crystal structure on Assaf’s key tableaux

Intertwine with weak Edelman–Greene insertion

Crystal for Schubert polynomials as Demazure truncation of crystalfor Stanley symmetric functions


Goal

Prove:


Strategy:

Impose Demazure crystal structure on Assaf’s key tableaux

Intertwine with weak Edelman–Greene insertion

Crystal for Schubert polynomials as Demazure truncation of crystalfor Stanley symmetric functions

Related work:

Lenart defined crystal operators on RC graphs

Reiner–Shimozono defined crystal-like operators on factorizedrow-frank words


Outline

1 Crystal: Schur functionsTableauxCrystal operators on tableaux

2 Crystal: Demazure charactersKey tableauxDemazure crystal on key tableaux

3 Crystal: Stanley symmetric functionsReduced factorizationsCrystal on reduced factorizations

4 Crystal: Schubert polynomialsReduced factorizations with cutoffCrystal on reduced factorizations with cutoff


Outline






Schur functions

Partition: λ

Young diagram of λ: array of left-justified cells with λi boxes in row i

(French convention)


Schur functions

Partition: λ


(French convention)

SSYTn(λ) = set of semi-standard Young tableaux of shape λ overalphabet {1, 2, . . . , n}


Schur functions

Partition: λ


(French convention)

SSYTn(λ) = set of semi-standard Young tableaux of shape λ overalphabet {1, 2, . . . , n}

Definition

Schur polynomial

sλ(x) = sλ(x1, . . . , xn) =∑

T∈SSYTn(λ)

xwt(T )11 · · · x

wt(T )nn


Semi-standard Young tableaux

Example

Semi-standard Young tableaux of shape (2, 1) over the alphabet {1, 2, 3}

32 3

31 3

21 3

21 2

32 2

31 2

31 1

21 1


Semi-standard Young tableaux

Example

Semi-standard Young tableaux of shape (2, 1) over the alphabet {1, 2, 3}

32 3

31 3

21 3

21 2

32 2

31 2

31 1

21 1

s(2,1)(x1, x2, x3) = x21 x2 + x21 x3 + x22 x3 + 2x1x2x3 + x1x22 + x1x

23 + x2x

23


Crystal structure on semi-standard Young tableaux

w word of length k in alphabet A = {1, 2, . . . , n}

Define

Mi(w , r) = wt(w1w2 · · ·wr )i−wt(w1w2 · · ·wr )i+1 (0 6 r 6 k , 1 6 i < n)




Define


Mi(w) = maxr>0

{Mi (w , r)}




Define


Mi(w) = maxr>0

{Mi (w , r)}

If Mi(w) > 0 and p leftmost occurrence of Mi(w) ⇒ wp = i

If q rightmost occurrence of Mi(w) ⇒ q = k or wq+1 = i + 1


Lowering operator

w(T ) column-reading word of T

Definition

Lowering operator fi on T :p be the smallest index such that Mi(w(T ), p) = Mi(w(T ))

if Mi(w(T )) 6 0, then fi(T ) = 0

else fi (T ) changes the entry in T corresponding to w(T )p = i to i +1


Lowering operator

w(T ) column-reading word of T

Definition

Lowering operator fi on T :p be the smallest index such that Mi(w(T ), p) = Mi(w(T ))

if Mi(w(T )) 6 0, then fi(T ) = 0

else fi (T ) changes the entry in T corresponding to w(T )p = i to i +1

Example

2 3 31 2 2 2 ❦2

2 3 31 2 2 ❦2 3

❦2 3 31 2 2 3 3

3 3 31 2 2 3 3 0

21323222 21323223 21323233 31323233

f2 f2 f2 f2


Raising operator

Definition

Raising operator ei on T :q largest index such that Mi(w(T ), q) = Mi(w(T ))

if q is length of w(T ), then ei (T ) = 0

else ei (T ) changes the entry in T corresponding to w(T )q+1 = i + 1to i


Raising operator

Definition

Raising operator ei on T :q largest index such that Mi(w(T ), q) = Mi(w(T ))

if q is length of w(T ), then ei (T ) = 0

else ei (T ) changes the entry in T corresponding to w(T )q+1 = i + 1to i

fi and ei are partial inverses


Crystal structure

The crystal B(2, 1) with edges f1 ↑, f2 ↑

32 3

31 3

32 2

21 3

31 2

21 2

31 1

21 1


Outline






Outline






Demazure character

Degree-preserving divided difference operators

πi f (x1, . . . , xn) = ∂i (xi f (x1, . . . , xn)) where ∂i =1− si

xi − xi+1

πw = πi1πi2 · · · πik if w = si1si2 · · · sik reduced expression


Demazure character

Degree-preserving divided difference operators

πi f (x1, . . . , xn) = ∂i (xi f (x1, . . . , xn)) where ∂i =1− si

xi − xi+1

πw = πi1πi2 · · · πik if w = si1si2 · · · sik reduced expression

Definition

Demazure character for composition a

κa(x) = πw

(xλ11 xλ2

2 · · · xλnn

)

where λ is partition rearrangement of a and w is shortest permutation thatsorts a to λ


Key tableaux

Weak composition: aKey diagram of a: array of left-justified cells with ai boxes in row i


Key tableaux


Definition

A key tableau is a filling of a key diagram such that:

columns have distinct entries

rows weakly decrease

if i is above k in same column with i < k , then ∃ j with i < j

immediately right of k


Key tableaux


Definition

A key tableau is a filling of a key diagram such that:

columns have distinct entries

rows weakly decrease

if i is above k in same column with i < k , then ∃ j with i < j

immediately right of k

Definition

A semi-standard key tableau is a key tableau in which no entry exceeds itsrow index.


Key tableaux

Example

Semi-standard key tableaux of shape (0, 2, 1)

21 1

31 1

32 1

32 2

12 2

13 3

is not allowed since 3 exceeds its row index 2


Demazure character/key polynomial

SSKT(a) = set of semi-standard key tableaux of shape a


Demazure character/key polynomial

SSKT(a) = set of semi-standard key tableaux of shape a

Theorem (Assaf 2017)

The key polynomial κa(x) is given by

κa(x) =∑

T∈SSKT(a)

xwt(T )11 · · · x

wt(T )nn


Key tableaux

Example

Semi-standard key tableaux of shape (0, 2, 1)

21 1

31 1

32 1

32 2

12 2

κ(0,2,1)(x) = x21 x2 + x21 x3 + x1x2x3 + x22 x3 + x1x22


Crystal operators on key tableaux


Define

mi (w , r) = wt(wr · · ·wk)i+1 − wt(wr · · ·wk)i (1 6 r 6 k , 1 6 i < n)




Define


mi(w) = maxr

{mi (w , r)}




Define


mi(w) = maxr

{mi (w , r)}

If mi (w) > 0 and q is rightmost occurrence of mi (w), then wq = i +1

If p leftmost occurrence of mi (w) ⇒ p = 1 or wp−1 = i


Raising operators

w(T ) column-reading word of T (right to left; reverse to tableaux)

Definition

Raising operator ei on T :q largest index such that mi (w(T ), q) = mi(w(T ))

if mi(w(T )) 6 0, then ei (T ) = 0

else ei (T ) changes all entries i + 1 weakly right of the entry in T

corresponding to w(T )q to i and all i ’s in the same columns as theseentries to i + 1’s


Raising operators

w(T ) column-reading word of T (right to left; reverse to tableaux)

Definition

Raising operator ei on T :q largest index such that mi (w(T ), q) = mi(w(T ))

if mi(w(T )) 6 0, then ei (T ) = 0

else ei (T ) changes all entries i + 1 weakly right of the entry in T

corresponding to w(T )q to i and all i ’s in the same columns as theseentries to i + 1’s

Example

3 1 12 2 2 2 ❦2

3 1 12 ❦2 2 2 1

3 2 2❦2 1 1 1 1

3 2 21 1 1 1 1 0

22121232 12121232 11212132 11212131

e1 e1 e1 e1


Lowering operators

Definition

Lowering operator fi on T :p smallest index such that mi(w(T ), p) = mi (w(T ))

if p = 1 or entry in T corresponding to wp lies in row i , thenfi(T ) = 0

else fi (T ) changes all entries i weakly right of the entry in T

corresponding to wp−1 to i + 1 and all i ’s in the same columns asthese entries to i ’s


Lowering operators

Definition

Lowering operator fi on T :p smallest index such that mi(w(T ), p) = mi (w(T ))

if p = 1 or entry in T corresponding to wp lies in row i , thenfi(T ) = 0

else fi (T ) changes all entries i weakly right of the entry in T

corresponding to wp−1 to i + 1 and all i ’s in the same columns asthese entries to i ’s

fi and ei are partial inverses


Crystal structure

Crystal SSKT(0, 2, 1) e1 ↑, e2 ↑

21 1

31 1

12 2

32 1

32 2


Crystal structure

Crystal SSKT(0, 2, 1) e1 ↑, e2 ↑ Crystal B(2, 1) f1 ↑, f2 ↑

21 1

31 1

12 2

32 1

32 2

32 3

31 3

32 2

21 3

31 2

21 2

31 1

21 1


Column sorting map

Definition

a weak composition of n, λ sorting of aColumn sorting map φ : SSKT(a) → SSYT(λ)


Column sorting map

Definition


Let cells fall vertically until no gaps between rows

Sort columns decreasingly bottom to top

Replace i 7→ n − i + 1.


Column sorting map

Definition





4 432 1


Column sorting map

Definition





4 432 1

fall−→ 4

3 42 1


Column sorting map

Definition





4 432 1

fall−→ 4

3 42 1

sort−→ 2

3 14 4


Column sorting map

Definition





4 432 1

fall−→ 4

3 42 1

sort−→ 2

3 14 4

replace−→ 3

2 41 1


Crystal structure

21 1

31 1

12 2

32 1

32 2

φ−→

32 3

31 3

32 2

21 3

31 2

21 2

31 1

21 1


Crystal structure

21 1

31 1

12 2

32 1

32 2

φ−→

32 3

31 3

32 2

21 3

31 2

21 2

31 1

21 1

T ∈ SSKT(a) ⇒ φ(ei (T )) = fn−i (φ(T )) if ei (T ) 6= 0


Demazure crystals

X ⊆ B(λ), define Di as

DiX = {b ∈ B(λ) | eki (b) ∈ X for some k > 0}


Demazure crystals



Definition (Demazure crystal)

w = si1si2 · · · sik ∈ Sn reduced expression, uλ ∈ B(λ) highest weight vector

Bw (λ) = Di1Di2 · · ·Dik{uλ}


Demazure crystals






Theorem (Littelmann (conjectured), Kashiwara (proven) 1993)

κa = chBw (λ) where w · a = λ


Demazure crystals






Theorem (Littelmann (conjectured), Kashiwara (proven) 1993)

κa = chBw (λ) where w · a = λ

Theorem (Assaf-S 2017)

SSKT(a) ∼= Bw (λ) where w · a = λ


Demazure crystal structure

Crystal SSKT(0, 2, 1) ∼= Bs1s2(2, 1)

21 1

31 1

12 2

32 1

32 2


Outline






Reduced factorizations

Definition

ρ reduced wordIncreasing factorization for ρ partitions ρ into factors which areincreasing

w permutationReduced factorization for w is an increasing factorization of a reducedword for w

RFℓ(w) = set of reduced factorizations of w into ℓ blocks


Reduced factorizations

Definition

ρ reduced wordIncreasing factorization for ρ partitions ρ into factors which areincreasing

w permutationReduced factorization for w is an increasing factorization of a reducedword for w

RFℓ(w) = set of reduced factorizations of w into ℓ blocks

Example

Reduced factorizations for w = 321 with reduced words 121 and 212 andℓ = 3:

()(12)(1) (12)()(1) (12)(1)() (1)(2)(1)

()(2)(12) (2)()(12) (2)(12)() (2)(1)(2)


Stanley symmetric functions

r reduced factorizationweight wt(r) is weak composition with ith part the number of letters inith block of r from the right

Example

wt((45)(3)(23)()) = (0, 2, 1, 2)




Example

wt((45)(3)(23)()) = (0, 2, 1, 2)

Definition

Stanley symmetric polynomial indexed by permutation w

Fw (x1, . . . , xℓ) =∑

r∈RFℓ(w−1)

xwt(r)




Example

wt((45)(3)(23)()) = (0, 2, 1, 2)

Definition

Stanley symmetric polynomial indexed by permutation w

Fw (x1, . . . , xℓ) =∑

r∈RFℓ(w−1)

xwt(r)

Example

F321(x1, x2, x3) = x21 x2 + x21x3 + x22x3 + 2x1x2x3 + x1x22 + x1x

23 + x2x

23

= s2,1(x1, x2, x3)


Crystal on reduced factorizations

Lowering operator fi on r :

Consider factors r i and r i+1

(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2





(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2

From large to small, pair x ∈ r i with smallest y ∈ r i+1 s.t. y > x

(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2





(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2


(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2

Remove smallest unpaired z ∈ r i and add z − t to r i+1

(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2

−→ (3 4 6 7)︸︷︷︸factor 3

(1 5 8 9)︸︷︷︸factor 2





(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2


(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2

Remove smallest unpaired z ∈ r i and add z − t to r i+1

(3 4 6)︸︷︷︸factor 3

(1 5 7 8 9)︸︷︷︸factor 2

−→ (3 4 6 7)︸︷︷︸factor 3

(1 5 8 9)︸︷︷︸factor 2

Example

(3 4 6)︸︷︷︸factor 3

(1 5 6 8 9)︸︷︷︸factor 2

−→ (3 4 5 6)︸︷︷︸factor 3

(1 5 8 9)︸︷︷︸factor 2


Stanley crystal

Theorem (Morse-S 2016)

RFℓ(w) with crystal operators fi and ei define an Aℓ−1-crystal structure


Stanley crystal



Corollary (Morse-S 2016)

Fw (x) =∑

r∈RFℓ(w−1)

ei r=0 ∀16i<ℓ

swt(r)(x)


Stanley crystal



Corollary (Morse-S 2016)

Fw (x) =∑

r∈RFℓ(w−1)

ei r=0 ∀16i<ℓ

swt(r)(x)

r 7→ (P(r),Q(r)) Edelman–Greene insertion


r ∈ RFℓ(w)

⇒ P(ei (r)) = P(r) and Q(ei (r)) = fℓ−i(Q(r)) if ei (r) 6= 0


Outline






Schubert polynomials

Definition

ρ = ρ1 . . . ρk reduced wordα = α1 . . . αk is ρ-compatible if α is weakly decreasing, αj 6 ρj , andαj > αj+1 whenever ρj > ρj+1


Schubert polynomials

Definition

ρ = ρ1 . . . ρk reduced wordα = α1 . . . αk is ρ-compatible if α is weakly decreasing, αj 6 ρj , andαj > αj+1 whenever ρj > ρj+1

R(w) set of reduced words of w

Theorem (Billey–Jockusch–Stanley 1993)

Schubert polynomial

Sw (x) =∑

ρ∈R(w−1)

∑

α∈RC(ρ)

xα


Reduced factorizations with cutoff

Example

Compatible sequences for reduced word 45323 for 153264:

45323

443224432144311442114321133211



Example


45323

443224432144311 (45)(3)()(23)442114321133211



Example


45323

44322 (45)(3)(23)()44321 (45)(3)(2)(3)44311 (45)(3)()(23)44211 (45)()(3)(23)43211 (4)(5)(3)(23)33211 ()(45)(3)(23)



Example


45323

44322 (45)(3)(23)()44321 (45)(3)(2)(3)44311 (45)(3)()(23)44211 (45)()(3)(23)43211 (4)(5)(3)(23)33211 ()(45)(3)(23)

Definition

Increasing factorization with cutoff is increasing factorization such thatfirst entry in block i from the right is at least i .



RFC(w) = set of reduced factorizations of w with cutoff

Proposition

Sw (x) =∑

r∈RFC(w−1)

xwt(r)




Proposition

Sw (x) =∑

r∈RFC(w−1)

xwt(r)

Observation: RFC(w) ⊆ RFn(w) for w ∈ Sn




Proposition

Sw (x) =∑

r∈RFC(w−1)

xwt(r)

Observation: RFC(w) ⊆ RFn(w) for w ∈ Sn

Fw (x) =∑

r∈RFℓ(w−1)

xwt(r)


Weak Edelman–Greene insertion

Example (Weak insertion)

(4)(5)(23)(2) 7→

4 4 5 4 5

2

4 5

2 3

4 532 3

4 4 3 4 3

2

4 3

2 2

4 312 2




(4)(5)(23)(2) 7→

4 4 5 4 5

2

4 5

2 3

4 532 3

4 4 3 4 3

2

4 3

2 2

4 312 2

r 7→ (P(r), Q(r)) weak Edelman–Greene insertion




(4)(5)(23)(2) 7→

4 4 5 4 5

2

4 5

2 3

4 532 3

4 4 3 4 3

2

4 3

2 2

4 312 2

r 7→ (P(r), Q(r)) weak Edelman–Greene insertion

Theorem (Assaf-S)

r ∈ RFC(w)⇒ P(ei (r)) = P(r) and Q(ei (r)) = ei (Q(r)) if ei (r) 6= 0


Demazure crystal for Schuberts

Theorem (Assaf-S)

RFC(w) ∼=⋃

r∈RFC(w)ei r=0 ∀16i<n

Bw(r)(wt(r))

where w(r) is the shortest permutation that sorts sh(P(r))


Demazure crystal for Schuberts

Theorem (Assaf-S)

RFC(w) ∼=⋃

r∈RFC(w)ei r=0 ∀16i<n

Bw(r)(wt(r))

where w(r) is the shortest permutation that sorts sh(P(r))

Corollary (Assaf-S)

Sw (x) =∑

r∈RFC(w−1)ei r=0 ∀16i<n

κsh(P(r))

(x)


Demazure crystal RFC(153264) e1 ր, e2 ↑, e3 տ

()(4)(35)(23)

()(45)(3)(23) (4)()(35)(23)

()(45)(23)(2) (4)(5)(3)(23) (4)(3)(5)(23)

(4)(5)(23)(2) (4)(3)(25)(3) (4)(35)()(23) (45)()(3)(23)

(4)(35)(2)(3) (45)()(23)(2) (45)(3)()(23)

(4)(35)(23)() (45)(3)(2)(3)

(45)(3)(23)()()(4)(3)(235)

()(4)(23)(25) (4)()(3)(235)

()(4)(235)(2) (4)()(23)(25) (4)(3)()(235)

(4)()(235)(2) (4)(3)(2)(35)

(4)(3)(23)(5)

(4)(3)(235)()


What’s next?

Type free generalization


What’s next?

Type free generalization

Analogue for affine Schur functions and affine Schubert polynomials


Thank you !

Documents

A Demazure crystals construction for Schubert polynomials · A Demazure crystals construction for Schubert polynomials Anne Schilling Department ofMathematics, UCDavis basedonjointworkwith