A uniform model for Kirillov-Reshetikhin crystals Cristian Lenart 1 Satoshi Naito 2 Daisuke Sagaki 3 Anne Schilling 4 Mark Shimozono 5 1 State University of New York, Albany 2 Tokyo Institute of Technology 3 University of Tsukuba 4 University of California, Davis 5 Virginia Tech FPSAC, June 26, 2013 Based on arXiv:1211.2042 and a forthcoming sequel.
Macdonald polynomials P ( x;q, 0) λ Ion ’03 affine ={types A−D−E} Demazure LNSSS characters Fourier− Littelmann ’06 Fourier− Schilling− (graded characters of) Shimozono ’07 Kirillov−Reshetikhin modules
Macdonald polynomials P ( x;q, 0) λ Ion ’03 affine ={types A−D−E} Demazure LNSSS characters Fourier− Littelmann ’06 Fourier− Schilling− (graded characters of) Shimozono ’07 Kirillov−Reshetikhin modules Theorem (LNSSS 2012) For all untwisted affine root systems, P λ ( x ; q , 0) = X λ ( x ; q ) , where X λ ( x ; q ) is the (graded) character of a tensor product of one-column Kirillov-Reshetikhin (KR) modules.
Summary P x;q,t Ram−Yip formula for ( ) λ uniform models for KR crystals X x; q P λ ( x;q, ( ) 0) = λ (the quantum alcove model)
Summary P x;q,t Ram−Yip formula for ( ) λ uniform models for KR crystals X x; q P λ ( x;q, ( ) 0) = λ (the quantum alcove model) computational applications R energy function combinatorial −matrix
Kashiwara’s crystals Colored directed graphs encoding certain representations V of the quantum group U q ( g ) as q → 0.
Kashiwara’s crystals Colored directed graphs encoding certain representations V of the quantum group U q ( g ) as q → 0. Kashiwara (crystal) operators are modified versions of the e i , � Chevalley generators (indexed by the simple roots): � f i .
Kashiwara’s crystals Colored directed graphs encoding certain representations V of the quantum group U q ( g ) as q → 0. Kashiwara (crystal) operators are modified versions of the e i , � Chevalley generators (indexed by the simple roots): � f i . Fact. V has a crystal basis B (vertices) = ⇒ in the limit q → 0 we have � f i , � e i : B → B ⊔ { 0 } , e i b ′ = b b → b ′ . � f i 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 � f 0 , � f 1 , . . . .
Kirillov–Reshetikhin (KR) crystals Correspond to certain finite -dimensional representations (not highest weight) of affine Lie algebras � g . The corresponding crystals have arrows � f 0 , � f 1 , . . . . Labeled by p × q rectangles, so they are denoted B p , q .
Kirillov–Reshetikhin (KR) crystals Correspond to certain finite -dimensional representations (not highest weight) of affine Lie algebras � g . The corresponding crystals have arrows � f 0 , � f 1 , . . . . Labeled by p × q rectangles, so they are denoted B p , q . We only consider column shapes B p , 1 .
Tensor products of KR crystals Definition. Given a composition p = ( p 1 , p 2 , . . . ), let B ⊗ p = B p 1 , 1 ⊗ B p 2 , 1 ⊗ . . . .
Tensor products of KR crystals Definition. Given a composition p = ( p 1 , p 2 , . . . ), let B ⊗ p = B p 1 , 1 ⊗ B p 2 , 1 ⊗ . . . . The crystal operators are defined on B ⊗ p by a tensor product rule.
Tensor products of KR crystals Definition. Given a composition p = ( p 1 , p 2 , . . . ), let B ⊗ p = B p 1 , 1 ⊗ B p 2 , 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 ( � sl n ) Fact. We have as classical crystals (without the 0-arrows): B p , 1 ≃ B ( ω p ) , where ω p = (1 , . . . , 1 , 0 , . . . , 0) = (1 p ) .
Models for KR crystals: type A (1) n − 1 ( � sl n ) Fact. We have as classical crystals (without the 0-arrows): B p , 1 ≃ B ( ω p ) , where ω p = (1 , . . . , 1 , 0 , . . . , 0) = (1 p ) . The vertices of this crystal are labeled by strictly increasing fillings of the Young diagram/column (1 p ) with 1 , . . . , n .
Models for KR crystals: type A (1) n − 1 ( � sl n ) Fact. We have as classical crystals (without the 0-arrows): B p , 1 ≃ B ( ω p ) , where ω p = (1 , . . . , 1 , 0 , . . . , 0) = (1 p ) . The vertices of this crystal are labeled by strictly increasing fillings of the Young diagram/column (1 p ) with 1 , . . . , n . The action of the crystal operators: � � � � f n − 1 f 1 f 2 f 0 1 − → 2 − → . . . n − 1 − − → n − → 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 A n − 1 – G 2 .
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, D B : 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, D B : 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, D B : 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, D B : 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 = R r .
Setup: finite root systems Root system Φ ⊂ V = R r . Reflections s α , α ∈ Φ .
Setup: finite root systems Root system Φ ⊂ V = R r . Reflections s α , α ∈ Φ . Example. Type A n − 1 . V = ( ε 1 + . . . + ε n ) ⊥ in R n = � ε 1 , . . . , ε n � ( r = n − 1). Φ = { α ij = ε i − ε j = ( i , j ) : 1 ≤ i � = j ≤ n } .
The Weyl group W = � s α : α ∈ Φ � .
The Weyl group W = � s α : α ∈ Φ � . Length function: ℓ ( w ) .
The Weyl group W = � s α : α ∈ Φ � . Length function: ℓ ( w ) . Example. Type A n − 1 . W = S n , s ε i − ε j = ( i , j ) is the transposition t ij .
The Weyl group W = � s α : α ∈ Φ � . Length function: ℓ ( w ) . Example. Type A n − 1 . W = S n , s ε i − ε j = ( i , j ) is the transposition t ij . The quantum Bruhat graph on W is the directed graph with labeled edges α w − → ws α ,
The Weyl group W = � s α : α ∈ Φ � . Length function: ℓ ( w ) . Example. Type A n − 1 . W = S n , s ε i − ε j = ( i , j ) is the transposition t ij . The quantum Bruhat graph on W is the directed graph with labeled edges α w − → ws α , where ℓ ( ws α ) = ℓ ( w ) + 1 (Bruhat graph) , or ℓ ( ws α ) = ℓ ( w ) − 2 ht ( α ∨ ) + 1 ( ht ( α ∨ ) = � ρ, α ∨ � ) .
The Weyl group W = � s α : α ∈ Φ � . Length function: ℓ ( w ) . Example. Type A n − 1 . W = S n , s ε i − ε j = ( i , j ) is the transposition t ij . The quantum Bruhat graph on W is the directed graph with labeled edges α w − → ws α , where ℓ ( ws α ) = ℓ ( w ) + 1 (Bruhat graph) , or ℓ ( ws α ) = ℓ ( w ) − 2 ht ( 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 S 3 : 321 α α 12 23 231 312 α α 13 13 α α 23 12 213 132 α α 12 23 123
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