Strobl,September9-12,2012 O.Azenhas,R.MamedeCMUC,UniversidadedeCoimbra KeypolynomialsoftypeCandcrystalgraphs

KC

Key polynomials of type C and crystal graphs

O. Azenhas, R. Mamede CMUC, Universidade de Coimbra

S´eminaire Lotharingien de Combinatoire 69

Strobl, September 9-12, 2012

KC

Outline

1 Introduction

2 Key polynomials of type A andgl_n-crystal graphs

3 Key polynomials of type C and sp_n-crystal graphs

KC Introduction

Key polynomials /Demazure characters

Key polynomials were introduced by Demazure for all Weyl groups (1974). They were studied combinatorially in the case of the symmetric group (type A) by Lascoux and Sch¨utzenberger (1988).

Ifgis a simple Lie algebra with Weyl group W, U_q(g) its quantum group, andV(λ) the integrable representation with highest weightλ anduλthe highest weight vector, for a givenw ∈W the Demazure module is defined to be

Vw(λ) :=Uq(g)^>0.u_w(λ),

and the Demazure character is the character ofVw(λ).

V(λ)−→crystal basis−→crystal graph (colored oriented graph).

KC Introduction

Key polynomials /Demazure characters

Key polynomials were introduced by Demazure for all Weyl groups (1974). They were studied combinatorially in the case of the symmetric group (type A) by Lascoux and Sch¨utzenberger (1988).

Ifgis a simple Lie algebra with Weyl group W, U_q(g) its quantum group, andV(λ) the integrable representation with highest weightλ andu_λthe highest weight vector, for a givenw ∈W the Demazure module is defined to be

Vw(λ) :=Uq(g)^>0.u_w(λ),

and the Demazure character is the character ofVw(λ).

V(λ)−→crystal basis−→crystal graph (colored oriented graph).

KC Introduction

Key polynomials /Demazure characters

It was conjectured by Littelmann (1991) and proved by Kashiwara (1993) that the intersection of a crystal basis ofVλ withVw(λ) is a crystal basis forVw(λ). The resulting subsetBw(λ)⊆B(λ) is called Demazure crystal.

The Demazure character is defined by the Demazure crystalBw(λ).

KC

Key polynomials of type A andgl_n-crystal graphs

Key polynomials of type A (L-S, 1988)

S_nis generated by the permutations s_i = (i i+ 1),i = 1, . . . ,n−1, The generatorss_i act on vectorsv= [v₁, . . . ,v_n]∈Nⁿby

siv = [v1, . . . ,vi+1,vi. . . ,vn], fori= 1, . . . ,n−1,

and induce an action of Sn onZ[x1,x2, ..,xn] by considering vectors v as exponents of monomialsx^v =x₁^v1x₂^v2· · ·x_n^vn.

Two families of Demazure operators: For i= 1, . . . ,n−1, πi,bπi :Z[x1,x2, ..,xn]−→Z[x1,x2, ..,xn],πbi=πi−1 where

πif = xif −xi+1f^si

x_i−x_i+1 andπbif = xi+1f −xi+1f^si

x_i−x_i+1 =πif −f. Two families of key polynomials: Forλa partition (at mostnparts) andw =s_iN. . .s_i2s_i1 a reduced decomposition inS_n, one defines the typeAkey polynomials indexed bywλ

κwλ(x) =πi_Nπi₂. . . πi₁x^λ and bκwλ(x) =bπi_Nbπi₂. . .bπi₁x^λ. Examples: monomialx^λandκωλ(x) =sλ(x) Schur polynomial.

KC

Key polynomials of type A andgl_n-crystal graphs

Key polynomials of type A (L-S, 1988)

S_nis generated by the permutations s_i = (i i+ 1),i = 1, . . . ,n−1, The generatorss_i act on vectorsv= [v₁, . . . ,v_n]∈Nⁿby

siv = [v1, . . . ,vi+1,vi. . . ,vn], fori= 1, . . . ,n−1,

and induce an action of Sn onZ[x1,x2, ..,xn] by considering vectors v as exponents of monomialsx^v =x₁^v1x₂^v2· · ·x_n^vn.

Two families of Demazure operators: For i= 1, . . . ,n−1, πi,bπi :Z[x1,x2, ..,xn]−→Z[x1,x2, ..,xn],πbi=πi−1 where

πif = xif −xi+1f^si

x_i−x_i+1 andπbif = xi+1f −xi+1f^si

x_i−x_i+1 =πif −f.

Two families of key polynomials: Forλa partition (at mostnparts) andw =s_iN. . .s_i2s_i1 a reduced decomposition inS_n, one defines the typeAkey polynomials indexed bywλ

κwλ(x) =πi_Nπi₂. . . πi₁x^λ and bκwλ(x) =bπi_Nbπi₂. . .bπi₁x^λ. Examples: monomialx^λandκωλ(x) =sλ(x) Schur polynomial.

KC

Key polynomials of type A andgl_n-crystal graphs

Key polynomials of type A (L-S, 1988)

S_nis generated by the permutations s_i = (i i+ 1),i = 1, . . . ,n−1, The generatorss_i act on vectorsv= [v₁, . . . ,v_n]∈Nⁿby

siv = [v1, . . . ,vi+1,vi. . . ,vn], fori= 1, . . . ,n−1,

and induce an action of Sn onZ[x1,x2, ..,xn] by considering vectors v as exponents of monomialsx^v =x₁^v1x₂^v2· · ·x_n^vn.

Two families of Demazure operators: For i= 1, . . . ,n−1, πi,bπi :Z[x1,x2, ..,xn]−→Z[x1,x2, ..,xn],πbi=πi−1 where

πif = xif −xi+1f^si

x_i−x_i+1 andπbif = xi+1f −xi+1f^si

x_i−x_i+1 =πif −f. Two families of key polynomials: Forλa partition (at mostnparts) andw =s_iN. . .s_i2s_i1 a reduced decomposition inS_n, one defines the typeAkey polynomials indexed bywλ

κwλ(x) =πi_Nπi₂. . . πi₁x^λ and bκwλ(x) =bπi_Nbπi₂. . .bπi₁x^λ. Examples: monomialx^λandκωλ(x) =sλ(x) Schur polynomial.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators and Demazure operators

Choose the alphabetAn={1<2< ... <n}. Consider a nonempty wordw on this alphabet and leti∈ {1, . . . ,n−1}.

Pick the subword consisting only of lettersi,i+ 1.

Encode

i−→( i+ 1−→).

Ignore successively all the factors ( ) to construct a new subword

ρ_i=)^r (^s. Definefi :A^∗_n−→A^∗_n∪ {0}:

Ifs= 0,fi(w) = 0. Ifs>0,fi(w) is obtained by changing the leftmost parentheses ) ofρi =)^r (^s into )

i−→i+ 1.

Ifr = 0,ei(w) = 0. Ifr >0,ei(w) is obtained by changing the rightmost parentheses ) ofρi=)^r (^s into (

i+ 1−→i.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal operators

f2: 212

((

)(

312

f2(212) = 312

)(

))

f₂²(212) = 313

e2(313) = 312 e2e2(312) = 212.

KC

Key polynomials of type A andgl_n-crystal graphs

gl

-crystal graph

Oriented colored graph with colors{1, . . . ,n−1}. An arrowa−→ⁱ biff fi(a) =b⇔ei(b) =a.

n= 3, λ= (210),B(210)gl₃-crystal 1 = 2 =

1 1 2

1 1

3 1 2

2

1 23 1 32

1 3

3 2 2

3

2 3 3

KC

Key polynomials of type A andgl_n-crystal graphs

Type A Demazure-crystal graph

The Demazure crystalBs₂s₁(210)⊂B(210) 1 = 2 =

1 1 2

1 1

3 1 2

2

1 2 3 1 3 2

1 3

3 2 2

3

2 3 3

fs₁(K) ={K,f1(K)}

: π1(x²¹⁰) =P

T∈f_s1(K)x^wt(T)

=x²¹⁰+x¹²⁰

fs₂s₁(K) ={f₂^m2f₁^m1(K) :m1,m2≥0}

={K,f2(K)} ∪ {f₁(K),f2f1(K),f₂²f1(K)}

: π2(x²¹⁰) +π2(x¹²⁰) =P

T∈f_s2s1(K)x^wt(T)

= (x²¹⁰+x²⁰¹) + (x¹²⁰+x¹¹¹+x¹⁰²)

κs₂s₁λ(x) =π2π1(x^λ) =P

T∈B_s2s1(210)x^wt(λ)

=bκλ+bκs₁λ+bκs₂λ+bκs₂s₁λ

=P

ν<s₂s₁bκνλ

KC

Key polynomials of type A andgl_n-crystal graphs

Type A Demazure-crystal graph

The Demazure crystalBs₂s₁(210)⊂B(210) 1 = 2 =

1 1 2

1 1

3 1 2

2

1 2 3 1 3 2

1 3

3 2 2

3

2 3 3

fs₁(K) ={K,f1(K)}

: π1(x²¹⁰) =P

T∈f_s1(K)x^wt(T)

=x²¹⁰+x¹²⁰

fs₂s₁(K) ={f₂^m2f₁^m1(K) :m1,m2≥0}

={K,f2(K)} ∪ {f₁(K),f2f1(K),f₂²f1(K)}

: π2(x²¹⁰) +π2(x¹²⁰) =P

T∈f_s2s1(K)x^wt(T)

= (x²¹⁰+x²⁰¹) + (x¹²⁰+x¹¹¹+x¹⁰²)

κs₂s₁λ(x) =π2π1(x^λ) =P

T∈B_s2s1(210)x^wt(λ)

=bκλ+bκs₁λ+bκs₂λ+bκs₂s₁λ

=P

ν<s₂s₁bκνλ

KC

Key polynomials of type A andgl_n-crystal graphs

Type A Demazure-crystal graph

The Demazure crystalBs₂s₁(210)⊂B(210) 1 = 2 =

1 1 2

1 1

3 1 2

2

1 2 3 1 3 2

1 3

3 2 2

3

2 3 3

fs₁(K) ={K,f1(K)}

: π1(x²¹⁰) =P

T∈f_s1(K)x^wt(T)

=x²¹⁰+x¹²⁰

fs₂s₁(K) ={f₂^m2f₁^m1(K) :m1,m2≥0}

={K,f₂(K)} ∪ {f₁(K),f2f1(K),f₂²f1(K)}

: π2(x²¹⁰) +π2(x¹²⁰) =P

T∈f_s2s1(K)x^wt(T)

= (x²¹⁰+x²⁰¹) + (x¹²⁰+x¹¹¹+x¹⁰²)

κs₂s₁λ(x) =π2π1(x^λ) =P

T∈B_s2s1(210)x^wt(λ)

=bκλ+bκs₁λ+bκs₂λ+bκs₂s₁λ

=P

ν<s₂s₁bκνλ

KC

Key polynomials of type A andgl_n-crystal graphs

Type A Demazure-crystal graph

The Demazure crystalBs₂s₁(210)⊂B(210) 1 = 2 =

1 1 2

1 1

3 1 2

2

1 2 3 1 3 2

1 3

3 2 2

3

2 3 3

fs₁(K) ={K,f1(K)}

: π1(x²¹⁰) =P

T∈f_s1(K)x^wt(T)

=x²¹⁰+x¹²⁰

fs₂s₁(K) ={f₂^m2f₁^m1(K) :m1,m2≥0}

={K,f₂(K)} ∪ {f₁(K),f2f1(K),f₂²f1(K)}

: π2(x²¹⁰) +π2(x¹²⁰) =P

T∈f_s2s1(K)x^wt(T)

= (x²¹⁰+x²⁰¹) + (x¹²⁰+x¹¹¹+x¹⁰²)

κs₂s₁λ(x) =π2π1(x^λ) =P

T∈B_s2s1(210)x^wt(λ)

=bκλ+bκs₁λ+bκs₂λ+bκs₂s₁λ

=P

ν<s₂s₁bκνλ

KC

Key polynomials of type A andgl_n-crystal graphs

Tableau criterion for Bruhat order on S

σ,µ∈Sn,v = (n,n−1, . . . ,2,1)∈Nⁿ

σ≥µiffkey(σv)≥key(µv)

key(v1,v2, . . . ,vn)=semistandard tableau of shape the decreasing rearrangement of (v1, . . . ,vn) whose firstvi columns contain the letteri

n= 4

key(s2s3s1s2(4,3,2,1)) =key(2,1,4,3), key(s2s3s1(4,3,2,1)) =key(3,1,4,2)

key(2,1,4,3) =

1 1 3 3

2 3 4 3 4 4

≥key(3,1,4,2) =

1 1 1 3

2 3 3 3 4 4 s2s3s1s2≥s2s3s1

KC

Key polynomials of type A andgl_n-crystal graphs

Characterisation of the tableaux in the gl

-Demazure crystal, L-S (1988)

key polynomial innvariablesx = (x1, . . . ,xn) κ_wλ(x) = X

T∈B_w(λ)

x^wtT = X

key(β)≤key(wλ)

bκ_β.

bκβ(x) = X

SSYTT wtT∈Nⁿ sh(T)=λ K⁺(T)=key(β)

x^wtT,

κwλ(x) = X

SSYTT wtT∈Nⁿ sh(T)=λ K⁺(T)≤key(wλ)

x^wtT.

KC

Key polynomials of type C andsp_n-crystal graphs

Hyperoctahedral group S

The hyperoctahedral groupS^B_n (2ⁿn! elements) is the Weyl group of the symplectic groupSp(2n,C). S^B_n is generated by the sign permutationssi= (i,i+ 1)(i,i+ 1),i = 1, . . . ,n−1 and sn= (n,n), which satisfy the relations:

1 s_i²= 1, i= 1, . . . ,n;

2 sisj =sjsi if|i−j| ≥2;

3 sisi+1si =si+1sisi+1, i= 1, . . . ,n−2;

4 s_n−1sns_n−1sn=sns_n−1sns_n−1.

[123¯4] =s₄, [1243] =s₃, [124¯3] =s₃s₄,

[12¯43] =s4s3, [12¯34] =s3s4s3, [12¯3¯4] =s4s3s4s3, [¯1¯2¯3¯4]

KC

Key polynomials of type C andsp_n-crystal graphs

S

The generatorss_i act on vectorsv= [v₁, . . . ,v_n]∈Zⁿ by s_iv = [v₁, . . . ,v_i−1,v_i+1,v_i. . . ,v_n], fori= 1, . . . ,n−1, and

snv= [v1, . . . ,v_n−1,−vn].

This induces an action ofS^B_n on the Laurent polynomials ring Z[x₁^±, . . . ,x_n^±] by considering vectors v∈Zⁿ as exponents of monomials x^v =x₁^v1x₂^v2· · ·x_n^vn.

KC

Key polynomials of type C andsp_n-crystal graphs

Type C Demazure operators

Two families of Demazure operators: For i= 1, . . . ,n, π^C_i ,bπ^C_i :Z[x^±,n]−→Z[x^±,n],πb^C_i =π^C_i −1 where

π^C_i f = x_if −x_i+1f^si xi−xi+1

and bπ^C_i f = x_i+1f −x_i+1f^si xi−xi+1

, i6=n and

π^C_nf = x_n²f −f^sn

x_n²−1 and πb^C_nf = f −f^sn x_n²−1.

Two families of key polynomials: Forλa partition (at mostnparts) andw =s_i^C

N. . .s_i^C

2s_i^C

1 reduced decomposition, one defines typeC key polynomials indexed bywλ

κ^C_wλ(x) =π_i^C

1πi₂. . . π_i^C

Nx^λ and bκ^C_wλ(x) =πb_i^C

1bπ_i^C

2. . .πb^C_i

nx^λ. Examples: monomialx^λandκ^C_ωλ(x) =spλ(x) symplectic Schur polynomial.

KC

Key polynomials of type C andsp_n-crystal graphs

Type C Demazure operators

Two families of Demazure operators: For i= 1, . . . ,n, π^C_i ,bπ^C_i :Z[x^±,n]−→Z[x^±,n],πb^C_i =π^C_i −1 where

π^C_i f = x_if −x_i+1f^si xi−xi+1

and bπ^C_i f = x_i+1f −x_i+1f^si xi−xi+1

, i6=n and

π^C_nf = x_n²f −f^sn

x_n²−1 and πb^C_nf = f −f^sn x_n²−1.

Two families of key polynomials: Forλa partition (at mostnparts) andw =s_i^C

N. . .s_i^C

2s_i^C

1 reduced decomposition, one defines typeC key polynomials indexed bywλ

κ^C_wλ(x) =π^C_i

1π_i2. . . π_i^C

Nx^λ and κb^C_wλ(x) =πb_i^C

1bπ_i^C

2. . .πb^C_i

nx^λ. Examples: monomialx^λandκ^C_ωλ(x) =sp_λ(x) symplectic Schur polynomial.

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic crystal operators

Choose the alphabetCn={1<2< ... <n<n¯< ... <¯2<¯1}. Consider a nonempty wordwon this alphabet and leti∈ {1, . . . ,n}.

Pick the subword consisting only of lettersi+ 1,¯i,i,i+ 1.

Encode

i+ 1,i−→+ ¯i,i+ 1−→ − Ignore all the successive factors +−to construct a new subword

ρi=−^r+^s Definefi:C_n^∗−→C_n^∗∪ {0}:

Ifs= 0,fi(w) = 0. Ifs>0,fi(w) is the word obtained by changing the left most symbol + ofρi=−^r+^sinto−where

i−→i+ 1, i+ 1−→¯i, and wheni=n, n−→n.¯

Ifr= 0,ei(w) = 0. Ifr>0,ei(w) is obtained by changing the rightmost symbol−ofρi =)^r(^sinto + where

i+ 1−→i ¯i−→i+ 1, and wheni=n, ¯n−→n.

KC

Key polynomials of type C andsp_n-crystal graphs

Example

ω= 12¯1¯1¯22121¯1¯2

+− − −+−+ +−+

Ignore all factors + −

− − +

− − −

¯2−→¯1 ω= (12)¯1¯1(¯22)(12)(1¯1)¯1.

KC

Key polynomials of type C andsp_n-crystal graphs

Example

ω= 12¯1¯1¯22121¯1¯2

+− − −+−+ +−+

Ignore all factors + −

− − + − − −

¯2−→¯1 ω= (12)¯1¯1(¯22)(12)(1¯1)¯1.

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic tableaux (De Concini, Kashiwara-Nakashima)

Admissible columns on the alphabet

C_n={1<2< ... <n<¯n< ... <¯2<¯1}. n= 6

P= 2 4 6

¯6

¯4

∅ 2 ∅ 4 ∅ 6

∅ ∅ ∅ ¯4 ∅ ¯6 Q= 2 3 4

¯4

¯2

∅ 2 3 4 ∅ ∅

∅ ¯2 ∅ ¯4 ∅ ∅

A column is an admissible column iff the diagram is such that there is a matching which sends each full slot to an empty slot to its left.

Splitting column form

P= 2 4 5

¯6

¯5

∅ 2 ∅ 4 5 ∅

∅ ∅ ∅ ∅ ¯5 ¯6 (`T,rT) = 2 2 3 4 4 5

¯6 ¯6

¯5 ¯3

A SSYT tableauT with admissible columnsT1,T2, . . . ,Tc is a symplectic tableau if the split column form of T is semistandard.

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic tableaux

P=

1 2 2 ¯1 4 4 ¯3

¯4 ¯2 ¯1

¯3

(`P,rP) =

1 1| 1 2| 2 2| ¯1 ¯1 2 4| 4 4| ¯3 ¯3|

¯4 ¯3| ¯2 ¯1| ¯1 ¯1|

¯3 ¯2|

wt(P) = (d₁, . . . ,d_n),d_i is the number of lettersi minus the number of letters ¯i,

wt(P) = (−1,1,−2,1).

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic tableaux

P=

1 2 2 ¯1 4 4 ¯3

¯4 ¯2 ¯1

¯3

(`P,rP) =

1 1| 1 2| 2 2| ¯1 ¯1 2 4| 4 4| ¯3 ¯3|

¯4 ¯3| ¯2 ¯1| ¯1 ¯1|

¯3 ¯2|

wt(P) = (d₁, . . . ,d_n),d_i is the number of lettersi minus the number of letters ¯i,

wt(P) = (−1,1,−2,1).

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic key tableaux

(v1,v2, . . . ,vn)∈Zⁿ, alphabet

Cn={1<2< ... <n<¯n< ... <¯2<¯1}.

key(v1,v2, . . . ,vn)=semistandard tableau of shape (|v1|, . . . ,|vn|)⁺ whose first|vi|columns contain the letteri ifvi≥0, and−i if v_i <0.

key(−1,0,3,−2,5,0,−3) =

3 3 3 5 5

5 5 5

¯7 ¯7 ¯7

¯4 ¯4

¯1

A column of a key either containsj or ¯j but not both. The columns of a key are admissible and its splitting form is a semistandard tableau.

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic crystal graph sp

1 1 2 1 2

2

1 1

¯2

1 ¯2 2

1 2

¯2

2 2

¯2 2 2

¯1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

: π2(x^¯12) =x^¯12+x^¯10+x^¯1¯2 π2(x⁰¹) =x⁰¹+x^0¯1 π2(x¹²) =x¹⁰+x^1¯1

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

: π2(x^¯12) =x^¯12+x^¯10+x^¯1¯2 π2(x⁰¹) =x⁰¹+x^0¯1 π2(x¹²) =x¹⁰+x^1¯1

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal of a sp

-crystal

Key polynomial of type C

κ^C_s

1s₂λ(x₁,x₂) = X

T∈Bs1s2(21)

x^wtT =x⁽²¹⁾+x^(2¯1)+x⁽¹²⁾+x⁽¹⁰⁾+x⁽⁰¹⁾x^(¯1,2)=

=x₁²x2+x₁²x₂⁻¹+x1x₂²+x1+x2+x₁⁻¹x₂².

wtT = (d1, . . . ,dn)∈Zⁿ, where di is the number of lettersi in T minus the number of letters ¯i in T.

KC

Key polynomials of type C andsp_n-crystal graphs

Tableau criterion for Bruhat order on S

σ,µ∈S^B_n,σ≤^C µ⇒B_σ⊆B_µ v = (n,n−1, . . . ,2,1)∈Nⁿ

σ≤^C µiffkey^C(σv)≤key^C(µv) n= 4, µ=s1s2s4s3s4s3≥^C σ=s2s3s4

key^C(µ(4321))≥key^C(σ(4321))

key^C(¯2,43¯1) =

2 2 2 2

3 3 3

¯4 ¯1

¯1

≥key^C(4¯132) =

1 1 1 1

3 3 3 4 4

¯2

KC

Key polynomials of type C andsp_n-crystal graphs

Symplectic Demazure crystal B

(21)

1 1 2 1 2

2 1 1

¯2

1 ¯2 2

1 2¯ 2

2 2¯ 2

2 2¯ 1 2 ¯2

¯2 2 ¯2

¯1

¯2 ¯2

¯1

¯2 ¯1

¯1 1 ¯2

¯2 1 ¯1

¯2 1 ¯1 2

2 ¯1

¯2

2 ¯1

¯1

: π2(x²¹) =x²¹+x^2¯1

: π1(x²¹) =x²¹+x¹² π1(x^2¯1) =x^2¯1+x¹⁰+x⁰¹+x^¯12

: π2(x^¯12) =x^¯12+x^¯10+x^¯1¯2 π2(x⁰¹) =x⁰¹+x^0¯1 π2(x¹²) =x¹⁰+x^1¯1