On relations between the classical and the Kazhdan-Lusztig representations of symmetric groups and associated Hecke algebras
Date
2005Source
Journal of Pure and Applied AlgebraVolume
203Issue
1-3Pages
133-144Google Scholar check
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Let H be the Hecke algebra of a Coxeter system (W, where W is a Weyl group of type An, over the ring of scalars A = Z[q1/2, q-1/2, where q is an indeterminate. We show that the Specht module Sλ, 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. © 2005 Elsevier B.V. All rights reserved.