A preorder-free construction of the Kazhdan-Lusztig representations - PowerPoint PPT Presentation

A preorder-free construction of the Kazhdan-Lusztig representations of Hecke algebras H n ( q ) of symmetric groups Charles Buehrle and Mark Skandera Department of Mathematics Lehigh University August 2010

The Hecke algebra, H n ( q ) 1 ¯ 1 2 ]-algebra generated by { � 2 , q C [ q T s i | 1 ≤ i ≤ n − 1 } with relations 1 ¯ 1 T 2 � 2 ) � T s i + � 2 − q s i = ( q T e , 1 ≤ i ≤ n − 1 T s i � � T s j � T s i = � T s j � T s i � T s j , | i − j | = 1 T s i � � T s j = � T s j � T s i , | i − j | ≥ 2 . The natural basis of H n ( q ) is the set { � T v = � T s i 1 · · · � T s i ℓ ( v ) | v ∈ S n } . Notice that H n (1) ∼ = C [ S n ]. For a v ∈ S n let P ( v ) , Q ( v ) be the tableaux obtained from Robinson-Schensted column insertion.

The Kazhdan-Lusztig basis of H n ( q ) In [Kazhdan and Lusztig, 1979] a certain basis of H n ( q ) is defined for each v ∈ S n to be � 1 2 ) ℓ ( v ) − ℓ ( u ) P u , v ( q ) � C ′ v = ( q T u , u ≤ v where P u , v ( q ) are the Kazhdan-Lusztig polynomials. Although, P u , v ( q ) ∈ N [ q ] there is no simple combinatorial description of the coefficients.

� � � � � � � Kazhdan-Lusztig preorders on H n ( q ) Kazhdan-Lusztig preorders help to construct representations. Right preorder T w = � u � ◮ v ⋖ R u if a v � = 0 in C ′ z ∈ S n a z C ′ z , for some w . ◮ The right preorder ≤ R is the transitive closure of ⋖ R . Example For S 3 the Hasse diagram of the right preorder is 123 � ���������������� � � �������� � � � � � � � � � � � � � � � � � � � � � � � � 312 213 231 132 � ���������������� � �������� � � � � � � � � � � � � � � � � � � � � � � � � 321

Kazhdan-Lusztig representations of H n ( q ) For λ ⊢ n choose a standard λ -tableau, T , and v such that Q ( v ) = T . Define K λ span { C ′ = u | Q ( u ) = T } span { C ′ u | u ≤ R v } / span { C ′ = u | u < R v } , def where u < R v means u ≤ R v �≤ R u . Matrix representations of H n ( q ) are obtained by right multiplication of � T s i on the “basis”. 1 ¯ 1 X λ 2 ]) d ) K : H n ( q ) → End (( C [ q 2 , q Example � � � � ¯ 1 1 − q 1 0 q 2 2 X (2 , 1) X (2 , 1) ( � ( � T s 1 ) = T s 2 ) = 1 ¯ 1 K K 0 q 1 − q 2 2

Kazhdan-Lusztig representation of H 3 ( q ) Choose λ = (2 , 1) and tableau T = 1 2 . 3 So we have that K λ = span { C ′ 213 , C ′ 312 } . ¯ 1 213 � C ′ 2 C ′ T s 1 = − q 213 1 312 � C ′ 2 C ′ 312 + C ′ 321 + C ′ = 213 . T s 1 q C ′ 321 is not in our spanning set! But, 321 < R 312. So we can ignore C ′ 321 due to the quotient. Thus � � ¯ 1 − q 1 2 X λ K ( � T s 1 ) = . 1 0 q 2

Quantum polynomial ring 1 ¯ 1 Define A ( n ; q ) = C [ q 2 , q 2 ] � x 1 , 1 , . . . , x n , n � , modulo x i ,ℓ x j , k = x j , k x i ,ℓ , 1 2 x i , k x i ,ℓ , x i ,ℓ x i , k = q 1 2 x i , k x j , k , x j , k x i , k = q 1 ¯ 1 2 − q 2 ) x i ,ℓ x j , k , x j ,ℓ x i , k = x i , k x j ,ℓ + ( q for 1 ≤ i < j ≤ n ,1 ≤ k < ℓ ≤ n . The relations can be remembered using the 2 × 2 submatrix � x i , k � x i ,ℓ x j , k x j ,ℓ

The immanant space and Kazhdan-Lusztig immanants Convenient monomial notation: x v , w = x v 1 , w 1 · · · x v n , w n . The immanant space span { x e , v | v ∈ S n } an n ! dimensional subspace of A ( n ; q ). In [Du, 1992] a dual canonical basis called Kazhdan-Lusztig immanants was defined for each u ∈ S n � 1 2 ) ℓ ( u ) − ℓ ( v ) P w 0 u , w 0 v ( q ) x e , v , Imm u ( x ) = ( − q v ≥ u where P w 0 u , w 0 v ( q ) are the inverse Kazhdan-Lusztig polynomials.

Generalized submatrices For n -element multisets of [ n ] L = ( ℓ (1) , . . . , ℓ ( n )) and M = ( m (1) , . . . , m ( n )) define   x ℓ (1) , m (1) · · · x ℓ (1) , m ( n )  . .  ... . . x L , M =  .  . . · · · x ℓ ( n ) , m (1) x ℓ ( n ) , m ( n ) Example L = (1 , 1 , 2) and M = (2 , 3 , 3)   x 1 , 2 x 1 , 3 x 1 , 3   . x L , M = x 1 , 2 x 1 , 3 x 1 , 3 x 2 , 2 x 2 , 3 x 2 , 3

Kazhdan-Lusztig representations of H n ( q ) , again For λ ⊢ n choose a standard λ -tableau, T , and v such that Q ( v ) = T . Define V λ = span { Imm u ( x ) | Q ( u ) = T } = span { Imm u ( x ) | u ≥ R v } / span { Imm u ( x ) | u > R v } . def H n ( q ) acts on V λ by � T u acting on the monomial basis { x e , v | v ∈ S n } . 1 ¯ 1 X λ 2 , q 2 ]) d ) V : H n ( q ) → End (( C [ q Theorem For any h ∈ H n ( q ) , X λ V ( h ) = X λ K ( h ) .

Kazhdan-Lusztig representation of H 3 ( q ) , again Choose λ = (2 , 1) and tableau T = 1 2 . 3 So we have that V λ = span { Imm 312 ( x ) , Imm 213 ( x ) } . ¯ 1 Imm 312 ( x ) � 2 Imm 312 ( x ) = − q T s 1 1 Imm 213 ( x ) � 2 Imm 213 ( x ) + Imm 123 ( x ) + Imm 312 ( x ) . T s 1 = q Imm 123 ( x ) is not in our spanning set! 213 < R 123. So we can ignore Imm 123 ( x ) due to the quotient. Thus � � ¯ 1 − q 1 2 V ( � X λ T s 1 ) = . 1 0 q 2

Vanishing of Kazhdan-Lusztig immanants Let L an n -element multiset of [ n ]. Theorem If ℓ ( i ) = ℓ ( i + 1) in L and s i u > u then Imm u ( x L , [ n ] ) = 0 . For n × n matrix A µ ( A ) = row multiplicity partition of A . Dominance order of partitions, λ � µ if � k i =1 λ i ≤ � k i =1 µ i , for all k . Theorem If sh ( u ) � µ ( x L , [ n ] ) then Imm u ( x L , [ n ] ) = 0 . These results are quantum analogues to results in [Rhoades and Skandera, 2009].

Quotient-free Kazhdan-Lusztig representations of H n ( q ) For λ ⊢ n , define the multiset L = (1 λ 1 , . . . , n λ n ). Define W λ = span { Imm u ( x L , [ n ] ) | Q ( u ) = T ( λ ) } . Matrix representations obtained by the action of H n ( q ) on basis of W λ . 1 ¯ 1 X λ 2 , q 2 ]) d ) W : H n ( q ) → End (( C [ q Theorem For any h ∈ H n ( q ) , X λ W ( h ) = X λ V ( h ) = X λ K ( h ) . This result is the H n ( q ) analog of a result in [B. and Skandera, 2010].

Quotient-free Kazhdan-Lusztig representation of H 3 ( q ) Choose λ = (2 , 1). We have that W λ = span { Imm 312 ( x 112 , 123 ) , Imm 213 ( x 112 , 123 ) } . ¯ 1 Imm 312 ( x 112 , 123 ) � 2 Imm 312 ( x 112 , 123 ) T s 1 = − q 1 Imm 213 ( x 112 , 123 ) � 2 Imm 213 ( x 112 , 123 ) + Imm 312 ( x 112 , 123 ) T s 1 = q + Imm 123 ( x 112 , 123 ) . sh (123) = (1 , 1 , 1) ≺ µ ( x 112 , 123 ) = (2 , 1). So Imm 123 ( x 112 , 123 ) = 0. Thus � � ¯ 1 − q 1 2 W ( � X λ T s 1 ) = . 1 0 q 2

Thank You B. and S. Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group. To appear in J. Pure Appl. Algebra , 2010. Michael Clausen Multivariate polynomials, standard tableaux, and representations of symmetric groups J. Symbolic Comput. , 11:483–522, 1991. J. Du Canonical bases for irreducible representations of quantum GL n . Bull. London Math. Soc. , 24(4):325-334, 1992. D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math. , 53:165–184, 1979. B. Rhoades and S. Bitableaux and the dual canonical basis of the polynomial ring. To appear in Adv. in Math. , 2009.