A uniform model for Kirillov-Reshetikhin crystals Cristian Lenart 1 - - PowerPoint PPT Presentation

A uniform model for Kirillov-Reshetikhin crystals

Cristian Lenart1 Satoshi Naito2 Daisuke Sagaki3 Anne Schilling4 Mark Shimozono5

1State University of New York, Albany 2Tokyo Institute of Technology 3University of Tsukuba 4University of California, Davis 5Virginia Tech

FPSAC, June 26, 2013 Based on arXiv:1211.2042 and a forthcoming sequel.

Littelmann ’06

characters Demazure (graded characters of) Kirillov−Reshetikhin modules 0) polynomials ( x;q, P Macdonald

={types A−D−E}

λ

LNSSS Ion ’03 Fourier− Schilling− Fourier− Shimozono ’07

affine

Littelmann ’06

characters Demazure (graded characters of) Kirillov−Reshetikhin modules 0) polynomials ( x;q, P Macdonald

={types A−D−E}

λ

LNSSS Ion ’03 Fourier− Schilling− Fourier− Shimozono ’07

affine

Theorem (LNSSS 2012)

For all untwisted affine root systems, Pλ(x; q, 0) = Xλ(x; q) , where Xλ(x; q) is the (graded) character of a tensor product of

• ne-column Kirillov-Reshetikhin (KR) modules.

Summary

Ram−Yip formula for 0) x;q, P λ( ) q = ) x;q,t (

λ

P uniform models for KR crystals (the quantum alcove model) ( X x;

λ

Summary

Ram−Yip formula for 0) x;q, P λ( ) q = ) x;q,t (

λ

P −matrix R uniform models for KR crystals (the quantum alcove model) computational applications energy function ( X x; combinatorial

λ

Kashiwara’s crystals

Colored directed graphs encoding certain representations V of the quantum group Uq(g) as q → 0.

Kashiwara’s crystals

Colored directed graphs encoding certain representations V of the quantum group Uq(g) as q → 0. Kashiwara (crystal) operators are modified versions of the Chevalley generators (indexed by the simple roots): ei, fi.

Kashiwara’s crystals

Colored directed graphs encoding certain representations V of the quantum group Uq(g) as q → 0. Kashiwara (crystal) operators are modified versions of the Chevalley generators (indexed by the simple roots): ei, fi.

• Fact. V has a crystal basis B (vertices) =

⇒ in the limit q → 0 we have

• fi,

ei : B → B ⊔ {0} ,

• fi b = b′

⇐ ⇒

• ei b′ = b

⇐ ⇒ b → b′ .

Kirillov–Reshetikhin (KR) crystals

Correspond to certain finite-dimensional representations (not highest weight) of affine Lie algebras g.

Kirillov–Reshetikhin (KR) crystals

Correspond to certain finite-dimensional representations (not highest weight) of affine Lie algebras g. The corresponding crystals have arrows f0, f1, . . ..

Kirillov–Reshetikhin (KR) crystals

Correspond to certain finite-dimensional representations (not highest weight) of affine Lie algebras g. The corresponding crystals have arrows f0, f1, . . .. Labeled by p × q rectangles, so they are denoted Bp,q.

Kirillov–Reshetikhin (KR) crystals

Correspond to certain finite-dimensional representations (not highest weight) of affine Lie algebras g. The corresponding crystals have arrows f0, f1, . . .. Labeled by p × q rectangles, so they are denoted Bp,q. We only consider column shapes Bp,1.

Tensor products of KR crystals

• Definition. Given a composition p = (p1, p2, . . .), let

B⊗p = Bp1,1 ⊗ Bp2,1 ⊗ . . . .

Tensor products of KR crystals

• Definition. Given a composition p = (p1, p2, . . .), let

B⊗p = Bp1,1 ⊗ Bp2,1 ⊗ . . . . The crystal operators are defined on B⊗p by a tensor product rule.

Tensor products of KR crystals

• Definition. Given a composition p = (p1, p2, . . .), let

B⊗p = Bp1,1 ⊗ Bp2,1 ⊗ . . . . The crystal operators are defined on B⊗p by a tensor product rule.

• Fact. B⊗p is connected (with the 0-arrows).

Models for KR crystals: type A(1)

n−1 (

sln)

• Fact. We have as classical crystals (without the 0-arrows):

Bp,1 ≃ B(ωp) , where ωp = (1, . . . , 1, 0, . . . , 0) = (1p) .

Models for KR crystals: type A(1)

n−1 (

sln)

• Fact. We have as classical crystals (without the 0-arrows):

Bp,1 ≃ B(ωp) , where ωp = (1, . . . , 1, 0, . . . , 0) = (1p) . The vertices of this crystal are labeled by strictly increasing fillings

• f the Young diagram/column (1p) with 1, . . . , n.

Models for KR crystals: type A(1)

n−1 (

sln)

• Fact. We have as classical crystals (without the 0-arrows):

Bp,1 ≃ B(ωp) , where ωp = (1, . . . , 1, 0, . . . , 0) = (1p) . The vertices of this crystal are labeled by strictly increasing fillings

• f the Young diagram/column (1p) with 1, . . . , n.

The action of the crystal operators: 1

• f1

− → 2

• f2

− → . . . n − 1

• fn−1

− − → n

• f0

− → 1 .

Models for KR crystals: types B(1)

n , C (1) n , D(1) n

• Fact. There are more involved type-specific models (based on

Kashiwara–Nakashima columns).

Models for KR crystals: types B(1)

n , C (1) n , D(1) n

• Fact. There are more involved type-specific models (based on

Kashiwara–Nakashima columns).

• Goal. Uniform model for all types A(1)

n−1 – G (1) 2 , based on the

corresponding finite root systems An−1 – G2.

The energy function

It originates in the theory of exactly solvable lattice models.

The energy function

It originates in the theory of exactly solvable lattice models. The energy function defines a grading on the classical components (no 0-arrows) of B = B⊗p (Schilling and Tingley).

The energy function

It originates in the theory of exactly solvable lattice models. The energy function defines a grading on the classical components (no 0-arrows) of B = B⊗p (Schilling and Tingley). More precisely, DB : B → Z≥0 satisfies the following conditions:

◮ it is constant on classical components (0-arrows removed);

The energy function

It originates in the theory of exactly solvable lattice models. The energy function defines a grading on the classical components (no 0-arrows) of B = B⊗p (Schilling and Tingley). More precisely, DB : B → Z≥0 satisfies the following conditions:

◮ it is constant on classical components (0-arrows removed); ◮ it decreases by 1 along certain 0-arrows.

The energy function

It originates in the theory of exactly solvable lattice models. The energy function defines a grading on the classical components (no 0-arrows) of B = B⊗p (Schilling and Tingley). More precisely, DB : B → Z≥0 satisfies the following conditions:

◮ it is constant on classical components (0-arrows removed); ◮ it decreases by 1 along certain 0-arrows.

• Goal. A more efficient uniform calculation, based only on the

combinatorial data associated with a crystal vertex

The energy function

It originates in the theory of exactly solvable lattice models. The energy function defines a grading on the classical components (no 0-arrows) of B = B⊗p (Schilling and Tingley). More precisely, DB : B → Z≥0 satisfies the following conditions:

◮ it is constant on classical components (0-arrows removed); ◮ it decreases by 1 along certain 0-arrows.

• Goal. A more efficient uniform calculation, based only on the

combinatorial data associated with a crystal vertex (type A: Lascoux–Sch¨ utzenberger charge statistic).

Setup: finite root systems

Root system Φ ⊂ V = Rr .

Setup: finite root systems

Root system Φ ⊂ V = Rr . Reflections sα, α ∈ Φ .

Setup: finite root systems

Root system Φ ⊂ V = Rr . Reflections sα, α ∈ Φ .

• Example. Type An−1.

V = (ε1 + . . . + εn)⊥ in Rn = ε1, . . . , εn (r = n − 1). Φ = {αij = εi − εj = (i, j) : 1 ≤ i = j ≤ n} .

The Weyl group

W = sα : α ∈ Φ .

The Weyl group

W = sα : α ∈ Φ . Length function: ℓ(w) .

The Weyl group

W = sα : α ∈ Φ . Length function: ℓ(w) .

• Example. Type An−1.

W = Sn , sεi−εj = (i, j) is the transposition tij .

The Weyl group

W = sα : α ∈ Φ . Length function: ℓ(w) .

• Example. Type An−1.

W = Sn , sεi−εj = (i, j) is the transposition tij . The quantum Bruhat graph on W is the directed graph with labeled edges w

α

− → wsα ,

The Weyl group

W = sα : α ∈ Φ . Length function: ℓ(w) .

• Example. Type An−1.

W = Sn , sεi−εj = (i, j) is the transposition tij . The quantum Bruhat graph on W is the directed graph with labeled edges w

α

− → wsα , where ℓ(wsα) = ℓ(w) + 1 (Bruhat graph) ,

• r

ℓ(wsα) = ℓ(w) − 2ht(α∨) + 1 (ht(α∨) = ρ, α∨) .

The Weyl group

W = sα : α ∈ Φ . Length function: ℓ(w) .

• Example. Type An−1.

W = Sn , sεi−εj = (i, j) is the transposition tij . The quantum Bruhat graph on W is the directed graph with labeled edges w

α

− → wsα , where ℓ(wsα) = ℓ(w) + 1 (Bruhat graph) ,

• r

ℓ(wsα) = ℓ(w) − 2ht(α∨) + 1 (ht(α∨) = ρ, α∨) . Comes from the multiplication of Schubert classes in the quantum cohomology of flag varieties QH∗(G/B) (Fulton and Woodward).

Bruhat graph for S3:

321

23 12 12 23 13 13 23 12

α α α α α α α α 123 132 213 231 312

Quantum Bruhat graph for S3:

321 α 13

23 12 12 23 13 13 23 12

α α α α α α α α 123 132 213 231 312

The quantum alcove model

• Definition. Given a dominant weight λ, we associate with it a

sequence of roots, called a λ-chain: Γ = (β1, . . . , βm) .

The quantum alcove model

• Definition. Given a dominant weight λ, we associate with it a

sequence of roots, called a λ-chain: Γ = (β1, . . . , βm) .

• Fact. A λ-chain corresponds to a sequence of alcoves.

The quantum alcove model

• Definition. Given a dominant weight λ, we associate with it a

sequence of roots, called a λ-chain: Γ = (β1, . . . , βm) .

• Fact. A λ-chain corresponds to a sequence of alcoves.

Let ri := sβi.

The quantum alcove model

• Definition. Given a dominant weight λ, we associate with it a

sequence of roots, called a λ-chain: Γ = (β1, . . . , βm) .

• Fact. A λ-chain corresponds to a sequence of alcoves.

Let ri := sβi. We consider subsets of positions in Γ J = (j1 < . . . < js) ⊆ {1, . . . , m} .

The quantum alcove model

• Definition. Given a dominant weight λ, we associate with it a

sequence of roots, called a λ-chain: Γ = (β1, . . . , βm) .

• Fact. A λ-chain corresponds to a sequence of alcoves.

Let ri := sβi. We consider subsets of positions in Γ J = (j1 < . . . < js) ⊆ {1, . . . , m} . We identify J with the chain in W w0 = Id, . . . , wi := rj1 . . . rji, . . . , ws = wend .

The quantum alcove model (cont.)

• Definition. A subset J = {j1 < j2 < . . . < js} is admissible if we

have a path in the quantum Bruhat graph Id = w0

βj1

− → w1

βj2

− → . . .

βjs

− → ws .

The quantum alcove model (cont.)

• Definition. A subset J = {j1 < j2 < . . . < js} is admissible if we

have a path in the quantum Bruhat graph Id = w0

βj1

− → w1

βj2

− → . . .

βjs

− → ws . A position ji is called a positive (resp. negative) folding if wi−1

βji

− → wi is an up (resp. down) step.

The quantum alcove model (cont.)

• Definition. A subset J = {j1 < j2 < . . . < js} is admissible if we

have a path in the quantum Bruhat graph Id = w0

βj1

− → w1

βj2

− → . . .

βjs

− → ws . A position ji is called a positive (resp. negative) folding if wi−1

βji

− → wi is an up (resp. down) step. Let J− := {ji : wi−1 > wi} .

The quantum alcove model (cont.)

• Definition. A subset J = {j1 < j2 < . . . < js} is admissible if we

have a path in the quantum Bruhat graph Id = w0

βj1

− → w1

βj2

− → . . .

βjs

− → ws . A position ji is called a positive (resp. negative) folding if wi−1

βji

− → wi is an up (resp. down) step. Let J− := {ji : wi−1 > wi} . Let A(Γ) = A(λ) be the collection of all admissible subsets.

The quantum alcove model (cont.)

• Definition. A subset J = {j1 < j2 < . . . < js} is admissible if we

have a path in the quantum Bruhat graph Id = w0

βj1

− → w1

βj2

− → . . .

βjs

− → ws . A position ji is called a positive (resp. negative) folding if wi−1

βji

− → wi is an up (resp. down) step. Let J− := {ji : wi−1 > wi} . Let A(Γ) = A(λ) be the collection of all admissible subsets.

• Construction. (L. and Lubovsky, generalization of L.-Postnikov,

Gaussent-Littelmann) Crystal operators f1, . . . , fr and f0 on A(λ).

The main result

• Theorem. (LNSSS) Given p = (p1, p2, . . .) and an arbitrary Lie

type, let λ = ωp1 + ωp2 + . . . .

The main result

• Theorem. (LNSSS) Given p = (p1, p2, . . .) and an arbitrary Lie

type, let λ = ωp1 + ωp2 + . . . . The (combinatorial) crystal A(λ) is isomorphic to the tensor product of KR crystals B⊗p.

Proof sketch

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum

Proof sketch

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Level 0 LS (Lakshmibai–Seshadri) paths: certain piecewise-linear paths with “directions” in Waf · λ.

Proof sketch

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Level 0 LS (Lakshmibai–Seshadri) paths: certain piecewise-linear paths with “directions” in Waf · λ. Projected level 0 LS paths: project to the finite weight lattice.

Proof sketch

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Level 0 LS (Lakshmibai–Seshadri) paths: certain piecewise-linear paths with “directions” in Waf · λ. Projected level 0 LS paths: project to the finite weight lattice.

• Fact. B⊗λ realized in terms of projected level 0 LS paths

(Naito-Sagaki ’03-’08).

Proof sketch

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Level 0 LS (Lakshmibai–Seshadri) paths: certain piecewise-linear paths with “directions” in Waf · λ. Projected level 0 LS paths: project to the finite weight lattice.

• Fact. B⊗λ realized in terms of projected level 0 LS paths

(Naito-Sagaki ’03-’08). Quantum LS paths: the “directions” in W · λ ≃ W /Wλ are related by paths in the parabolic quantum Bruhat graph QB(W /Wλ).

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Ingredients for (1): we lift QB(W /Wλ) to

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Ingredients for (1): we lift QB(W /Wλ) to

◮ Littelmann’s poset of level 0 weights Waf · λ

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Ingredients for (1): we lift QB(W /Wλ) to

◮ Littelmann’s poset of level 0 weights Waf · λ =

⇒ description

• f covers;

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Ingredients for (1): we lift QB(W /Wλ) to

◮ Littelmann’s poset of level 0 weights Waf · λ =

⇒ description

• f covers;

◮ the Bruhat order on Waf (parabolic version of “quantum to

affine”, cf. Peterson ’97, Lam–Shimozono ’10).

Ingredients in the proof

Projected level 0 LS paths = (2) (1) LS paths Quantum alcove model Quantum Ingredients for (1): we lift QB(W /Wλ) to

◮ Littelmann’s poset of level 0 weights Waf · λ =

⇒ description

• f covers;

◮ the Bruhat order on Waf (parabolic version of “quantum to

affine”, cf. Peterson ’97, Lam–Shimozono ’10). Ingredients for (2):

◮ we study various other properties of the parabolic quantum

Bruhat graph.

Example in type A2. p = (1, 2, 2, 1) = ; λ = ω1 + ω2 + ω2 + ω1 = (4, 2, 0).

Example in type A2. p = (1, 2, 2, 1) = ; λ = ω1 + ω2 + ω2 + ω1 = (4, 2, 0). A λ-chain as a concatenation of ω1-, ω2-, ω2-, and ω1-chains: Γ = ( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) .

• Example. Let J = {1, 2, 3, 6, 7, 8}.

( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) .

• Example. Let J = {1, 2, 3, 6, 7, 8}.

( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) . Claim: J is admissible. Indeed, the corresponding path in the quantum Bruhat graph is 1 2 3 < → 2 1 3 < → 3 1 2 | 3 1 2 < → 3 2 1 | 3 2 1 > → 1 2 3 | 1 2 3 < → 2 1 3 < → 3 1 2 .

• Example. Let J = {1, 2, 3, 6, 7, 8}.

( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) . Claim: J is admissible. Indeed, the corresponding path in the quantum Bruhat graph is 1 2 3 < → 2 1 3 < → 3 1 2 | 3 1 2 < → 3 2 1 | 3 2 1 > → 1 2 3 | 1 2 3 < → 2 1 3 < → 3 1 2 . The corresponding element in B⊗p = B1,1 ⊗ B2,1 ⊗ B2,1 ⊗ B1,1 represented via column-strict fillings: 3 ⊗ 2 3 ⊗ 1 2 ⊗ 3 .

The energy function in arbitrary Lie type

• Definition. Given the λ-chain

Γ = (β1, . . . , βm) , define the height sequence (h1, . . . , hm) by hi := #{j ≥ i : βj = βi} .

The energy function in arbitrary Lie type

• Definition. Given the λ-chain

Γ = (β1, . . . , βm) , define the height sequence (h1, . . . , hm) by hi := #{j ≥ i : βj = βi} . Then, for J ∈ A(λ), define the statistic height(J) :=

• j∈J−

hj .

The energy function in arbitrary Lie type

• Definition. Given the λ-chain

Γ = (β1, . . . , βm) , define the height sequence (h1, . . . , hm) by hi := #{j ≥ i : βj = βi} . Then, for J ∈ A(λ), define the statistic height(J) :=

• j∈J−

hj .

• Theorem. (LNSSS) Given J ∈ A(λ), which is identified with B⊗p,

we have DB(J) = −height(J) .

• Example. Consider the running example: λ = ω1 + ω2 + ω2 + ω1 in

type A2.

• Example. Consider the running example: λ = ω1 + ω2 + ω2 + ω1 in

type A2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} ∈ A(Γ): Γ = ( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) , (hi) = ( 2, 4 | 2, 3 | 1, 2 | 1, 1 ) .

• Example. Consider the running example: λ = ω1 + ω2 + ω2 + ω1 in

type A2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} ∈ A(Γ): Γ = ( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) , (hi) = ( 2, 4 | 2, 3 | 1, 2 | 1, 1 ) . We have height(J) = 2 .

• Example. Consider the running example: λ = ω1 + ω2 + ω2 + ω1 in

type A2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} ∈ A(Γ): Γ = ( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) , (hi) = ( 2, 4 | 2, 3 | 1, 2 | 1, 1 ) . We have height(J) = 2 .

• Remarks. (1) In type A, the height statistic translates into the

Lascoux–Sch¨ utzenberger charge statistic on Young tableaux (L.).

• Example. Consider the running example: λ = ω1 + ω2 + ω2 + ω1 in

type A2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} ∈ A(Γ): Γ = ( (1, 2), (1, 3) | (2, 3), (1, 3) | (2, 3), (1, 3) | (1, 2), (1, 3) ) , (hi) = ( 2, 4 | 2, 3 | 1, 2 | 1, 1 ) . We have height(J) = 2 .

• Remarks. (1) In type A, the height statistic translates into the

Lascoux–Sch¨ utzenberger charge statistic on Young tableaux (L.). (2) A similar charge statistic was defined in type C (L. and Schilling), and one is being developed in type B (Briggs and L.).