Existence and combinatorial model for KirillovReshetikhin crystals - PowerPoint PPT Presentation

Existence and combinatorial model for Kirillov–Reshetikhin crystals Anne Schilling Department of Mathematics University of California at Davis Aarhus June 28, 2007 – p. 1/ ?

References This talk is based on the following papers: • A. Schilling, Combinatorial structure of Kirillov–Reshetikhin crystals of type D (1) n , B (1) n , A (2) 2 n − 1 , preprint arXiv:0704.2046[math.QA] • M. Okado, A. Schilling, Existence of Kirillov–Reshetikhin crystals for nonexceptional types , preprint arXiv:0706.2224[math.QA] – p. 2/ ?

Quantum algebras Drinfeld and Jimbo ∼ 1984: independently introduced quantum groups U q ( g ) Kashiwara ∼ 1990: crystal bases, bases for U q ( g ) -modules as q → 0 combinatorial approach Lusztig ∼ 1990: canonical bases geometric approach – p. 3/ ?

Applications in... representation theory ❀ tensor product decomposition solvable lattice models ❀ one point functions conformal field theory ❀ characters number theory ❀ modular forms Bethe Ansatz ❀ fermionic formulas combinatorics ❀ tableaux combinatorics topological invariant theory ❀ knots and links – p. 4/ ?

Motivation • Crystal bases are combinatorial bases for U q ( g ) as q → 0 • Affine finite crystals: • appear in 1d sums of exactly solvable lattice models • path realization of integrable highest weight U q ( g ) -modules • fermionic formulas • Irreducible finite-dimensional U q ( g ) -modules classified by Chari-Pressley via Drinfeld polynomials – p. 5/ ?

Motivation • Kirillov-Reshetikhin modules W ( r ) form special s subset Conjecture [HKOTY] W ( r ) has a crystal basis B r,s s – p. 6/ ?

Motivation • Kirillov-Reshetikhin modules W ( r ) form special s subset Conjecture [HKOTY] W ( r ) has a crystal basis B r,s s AIM: • prove this conjecture for g of nonexceptional type B r,s for types • provide a combinatorial crystal ˜ D (1) n , B (1) n , A (2) 2 n − 1 • prove that B r,s ∼ = ˜ B r,s – p. 6/ ?

Motivation Solvable Lattice Models Bethe CTM Ansatz Bijection Rigged Highest Weight Configurations Crystals 1988 Identity for Kostka polynomials Kerov, Kirillov, Reshetikhin X = M conjecture of HKOTTY 2001 – p. 7/ ?

Outline I. Motivation II. Existence of KR crystals B r,s for nonexceptional types • Definition of KR modules • Criterion for existence B r,s of type D (1) III. Combinatorial KR crystals ˜ n , B (1) n , A (2) 2 n − 1 • Dynkin diagram automorphisms • Classical crystal structure • Affine crystal structure IV. MuPAD-Combinat implementation V. Outlook and open problems – p. 8/ ?

II. Existence of KR crystals B r,s for nonexceptional types – p. 9/ ?

Quantum affine algebras g symmetrizable affine Kac–Moody algebra U q ( g ) quantum affine algebra associated to g : associative algebra over Q ( q ) with 1 generated by e i , f i , q h for i ∈ I , h ∈ P ∗ { α i } i ∈ I simple roots, { h i } i ∈ I simple coroots c canonical central element, δ generator of null roots P = � i Z Λ i ⊕ Z δ weight lattice A subring of Q ( q ) of rational functions without poles at q = 0 A Z = { f ( q ) /g ( q ) | f ( q ) , g ( q ) ∈ Z [ q ] , g (0) = 1 } K Z = A Z [ q − 1 ] – p. 10/ ?

Prepolarization Let M be a U q ( g ) -module. A symmetric bilinear form ( , ) : M ⊗ Q ( q ) M → Q ( q ) is called prepolarization if ( q h u, v ) = ( u, q h v ) ( e i u, v ) = ( u, q − 1 i t − 1 i f i v ) ( f i u, v ) = ( u, q − 1 i t i e i v ) with q i = q ( α i ,α i ) / 2 , t i = q h i i . A prepolarization is called polarization if it is positive definite using the order f − g ∈ ∪ n ∈ Z { q n ( a + qA ) | a > 0 } f > g iff – p. 11/ ?

Criterion for existence M finite-dimensional integrable U ′ q ( g ) -module ( , ) prepolarization on M M K Z submodule of M such that ( M K Z , M K Z ) ⊂ K Z λ 1 , . . . , λ m ∈ P + Assumptions A : 1. dim M λ k ≤ � m j =1 dim V ( λ j ) λ k 2. There exist u j ∈ ( M K Z ) λ j such that ( u j , u k ) ∈ δ j,k + qA ( e i u j , e i u j ) ∈ qq − 2(1+ � h i ,λ j � ) A i – p. 12/ ?

Criterion for existence If Assumption A holds: Theorem: [KMN 2 ] (i) ( , ) is a polarization on M (ii) M ∼ = � j V ( λ j ) as U q ( g 0 ) -modules (iii) ( L, B ) is a crystal pseudobase of M , where L = { u ∈ M | ( u, u ) ∈ A } B = { b ∈ M K Z ∩ L/M K Z ∩ qL | ( b, b ) 0 = 1 } ( , ) 0 is Q -valued symmetric bilinear form on L/qL induced by ( , ) . – p. 13/ ?

KR modules Chari-Pressley classified all irreducible finite-dimensional affine U q ( g ) -modules via Drinfeld polynomials. KR modules W ( r ) ( s ∈ Z > 0 , r = 1 , . . . , n ) correspond s to the Drinfeld polynomials � (1 − a r q 1 − s ) · · · (1 − a r q s − 1 u ) j = r r r P j ( u ) = j � = r 1 for some a r ∈ Q ( q ) – p. 14/ ?

Construction of KR modules V ( λ ) extremal weight module level 0 fundamental weight ̟ i = Λ i − � c, Λ i � Λ 0 Define U ′ q ( g ) -module W ( ̟ i ) as W ( ̟ i ) = V ( ̟ i ) / ( z i − 1) V ( ̟ i ) where z i is a U ′ q ( g ) -module automorphism of V ( ̟ i ) of weight d i δ u ̟ i �→ u ̟ i + d i δ d i = max { 1 , ( α i , α i ) / 2 } W ( r ) can be obtained by from W ( ̟ r ) by the fusion s construction – p. 15/ ?

Existence Theorem [Okado,S.] W ( r ) has a crystal basis B r,s . s Assumption 1. follows from recent work by Nakajima and Hernandez on characters of KR-modules Assumption 2. follows by finding appropriate λ j and explicitly calculating the prepolarization in the cases : D (1) n , B (1) n , A (2) • Case 2 n − 1 : C (1) • Case n : A (2) 2 n , D (2) • Case n +1 – p. 16/ ?

Existence Theorem [Okado,S.] W ( r ) has a crystal basis B r,s . s Remark: [KMN 2 ] proved the existence of B r,s for type A (1) n and for other types for special r, s . – p. 16/ ?

B r,s of type D (1) n , B (1) III. Combinatorial KR crystals ˜ n , A (2) 2 n − 1 – p. 17/ ?

Axiomatic Crystals A U q ( g ) -crystal is a nonempty set B with maps wt: B → P e i , f i : B → B ∪ {∅} for all i ∈ I satisfying f i ( b ) = b ′ ⇔ e i ( b ′ ) = b if b, b ′ ∈ B wt( f i ( b )) = wt( b ) − α i if f i ( b ) ∈ B � h i , wt( b ) � = ϕ i ( b ) − ε i ( b ) b i b’ for b ′ = f i ( b ) Write r r ✲ – p. 18/ ?

KR crystals g affine Kac–Moody algebra W ( r ) KR module indexed by r ∈ { 1 , . . . , n } , s ≥ 1 s ❀ finite-dimensional U ′ q ( g ) -module Chari proved � ∼ W ( r ) V (Λ) as U q ( g 0 ) -module = s Λ – p. 19/ ?

KR crystals g affine Kac–Moody algebra W ( r ) KR module indexed by r ∈ { 1 , . . . , n } , s ≥ 1 s ❀ finite-dimensional U ′ q ( g ) -module Chari proved � ∼ W ( r ) V (Λ) as U q ( g 0 ) -module = s Λ g of type A (1) n ⇒ g 0 of type A n   �   ∼ W ( r ) = V r   s � �� �  s – p. 19/ ?

KR crystals g affine Kac–Moody algebra W ( r ) KR module indexed by r ∈ { 1 , . . . , n } , s ≥ 1 s ❀ finite-dimensional U ′ q ( g ) -module Chari proved � ∼ W ( r ) V (Λ) as U q ( g 0 ) -module = s Λ g of type D (1) n , B (1) n , A (2) 2 n − 1 ⇒ g 0 of type D n , B n , C n r with vertical dominos sum over removed � �� � – p. 19/ ? s

Example Type D (1) n , B (1) n , A (2) 2 n − 1 ∼ W (4) ) ⊕ W ( ) ⊕ W ( = W ( ) 2 ⊕ W ( ) ⊕ W ( ) ⊕ W ( ∅ ) – p. 20/ ?

Dynkin automorphism Type A (1) n : KMN 2 proved existence of crystals B r,s for W r,s Shimozono gave the combinatorial structure of B r,s using σ · · · n n-1 � � � A (1) n � � 0 � � � · · · 1 2 – p. 21/ ?

Dynkin automorphism Type A (1) n : KMN 2 proved existence of crystals B r,s for W r,s Shimozono gave the combinatorial structure of B r,s using σ · · · n n-1 � � � A (1) n � � 0 � � � · · · 1 2 e 0 = σ − 1 ◦ e 1 ◦ σ f 0 = σ − 1 ◦ f 1 ◦ σ – p. 21/ ?

Dynkin automorphism Type D (1) n , B (1) n , A (2) 2 n − 1 : n − 1 0 � � . . . n − 2 3 2 Type D (1) σ n : � � � � � n 1 � � 0 � . . . n n − 1 2 3 Type B (1) σ n : � � � � � � 1 � 0 � . . . n n − 1 3 2 Type A (2) σ 2 n − 1 : � � � � � � 1 � e 0 = σ ◦ e 1 ◦ σ f 0 = σ ◦ f 1 ◦ σ and – p. 22/ ?

Crystals B 1 , 1 0 D (1) n-1 n n n 1 2 n-2 n-2 2 1 · · · · · · 1 2 n-1 n-1 2 1 n n n-1 0 0 B (1) n 1 2 n-1 n n n-1 2 1 · · · · · · 1 2 n 0 n 2 1 0 0 A (2) 2 n − 1 1 2 n-1 n n-1 2 1 · · · · · · n 1 2 n 2 1 0 – p. 23/ ?

Classical decomposition By construction � B r,s ∼ B (Λ) = Λ as X n = D n , B n , C n crystals ⇒ crystal arrows f i , e i are fixed for i = 1 , 2 , . . . , n – p. 24/ ?

Classical crystal ) ⊗| Λ | B (Λ) ⊂ B ( highest weight 4 3 2 2 2 1 1 1 �→ 4 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 f i , e i for i = 1 , 2 , . . . , n act by tensor product rule b ⊗ b ′ − − − −− + + + + + ++ � �� � � �� � ���� � �� � ϕ i ( b ) ε i ( b ) ϕ i ( b ′ ) ε i ( b ′ ) – p. 25/ ?