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 , 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 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.

RSK bases and Kazhdan-Lusztig cells

“…They are about the determinant, denoted det(λ), of the matrix of a certain bilinear form, the Dipper-James form, on the Specht module S λ , computed with respect to the "standard basis" of S λ ; while G(λ) has to do with multiplication of Kazhdan-Lusztig basis elements. The connection between det(λ) and det G(λ) is established in §9 (see Equation (9.2)) using results of [24].…”

“…The justification for our modification of the standard definition of RSKcorrespondence is that it was the simplest way we could think up of reconciling the notational conflict among the two sets of papers upon which we rely: [6,24] and [15]. Permutations act on the right in the former -a convention which we too follow-but on the left in the latter.…”

“…In particular, to pass from our notation to that of [20], [6], or [24], we need to replace v by q 1/2 and T w by q − (w)/2 T w .…”

“…But in fact we will recall in some detail in §8.3 the following more general fact from [6,24]: Specht modules can be defined over the Hecke algebra H and are isomorphic to the corresponding right cell modules.…”

Bitableaux and Zero Sets of Dual Canonical Basis Elements

“…It is therefore natural to hope that suitable orderings of the two bases leads to a transition matrix having a particularly nice form. This hope seems even more natural if one considers the results in [22], [32], [33], [40], which relate bases constructed from Kazhdan-Lusztig polynomials to others constructed from Young tableaux. Indeed we will show in Theorem 5.8 -Corollary 5.12 that the transition matrix relating the dual canonical basis and bitableau basis is an infinite direct sum of unitriangular transition matrices, each of which corresponds to a component…”