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.

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A complete determination of the irreducible modules of specialized Hecke algebras of type F , with respect to specializations with equal parameters, was 4 Ž . obtained by M. Geck and K. Lux 1991, Manuscripta Math. 70, 285᎐306 . A similar determination for specializations with¨s u 2 and¨s u 4 was obtained by K.Ž . Bremke 1994, Manuscripta Math. 83, 331᎐346 . In this paper, we determine the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3, obtaining en route decompositions of the generic irreducible modules under such specializations. We find that the decomposition matrices may be expressed in lower uni-triangular form in all these cases.

Abstract.Given an involution z in W , where W is the symmetric group of degree n, we study the relation between the subsystems of a root system for W corresponding to certain decreasing subsequences of z and the two-sided Kazhdan-Lusztig cell of W containing z. We study the finite symmetric group W which is a finite Coxeter group of type A. In this case, each left cell and each right cell contains exactly one involution, and each two-sided cell contains exactly one involution of a special form-a standard parabolic involution-which we describe below. The parabolic involutions form a larger collection of involutions, each of which may be associated with the standard parabolic involution in the two-sided cell containing it in a simple combinatorial way. The Robinson-Schensted-Knuth process provides a combinatorial technique for identifying the standard parabolic involution in the same two-sided cell as a given involution. Our aim is to provide a simpler combinatorial technique for carrying out this identification for a large proportion of the involutions. Not all involutions will be covered by the technique, since they must satisfy a certain length restriction which is not satisfied by all involutions. We have computed the proportion of involutions failing this restriction for symmetric groups of degree ≤12 and found it to be <0.007. Similarly, the proportion of involutions not covered by our combinatorial technique is <0.016.Moreover, if is a system of roots for W , we show that in order to identify the twosided cell of W containing an involution z ∈ W , it suffices to consider the realizations of z as the longest element of a Young subgroup W ( ) with respect to a simple generating

For a composition [Formula: see text] of [Formula: see text] we consider the Kazhdan–Lusztig cell in the symmetric group [Formula: see text] containing the longest element of the standard parabolic subgroup of [Formula: see text] associated to [Formula: see text]. In this paper, we extend some of the ideas and results in [Beiträge zur Algebra und Geometrie, 59(3) (2018) 523–547]. In particular, by introducing the notion of an ordered [Formula: see text]-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.