On relations between the classical and the Kazhdan–Lusztig representations of symmetric groups and associated Hecke algebras

Abstract

Let H be the Hecke algebra of a Coxeter system ( W , S ) , where W is a Weyl group of type A n , over the ring of scalars A = Z [ q 1 / 2 , q - 1 / 2 ] , where q is an indeterminate. We show that the Specht module S λ , as defined by Dipper and James [Proc. London Math. Soc. 52(3) (1986) 20–52], is naturally isomorphic over A to the cell module of Kazhdan and Lusztig [Invent. Math. 53 (1979) 165–184] associated with the cell containing the longest element of a parabolic subgroup W J for appropriate J ⊆ S . We give the association between J and λ explicitly. We introduce notions of the T-basis and C-basis of the Specht module and show that these bases are related by an invertible triangular matrix over A. We point out the connection with the work of Garsia and McLarnan [Adv. Math. 69 (1988) 32–92] concerning the corresponding representations of the symmetric group.