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On relations between the classical and the Kazhdan-Lusztig representations of symmetric groups and associated Hecke algebras. (English) Zbl 1090.20006

The authors investigate the relation between the classical and the Kazhdan-Lusztig representations of the symmetric groups and associated Hecke algebras. 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 \(A=\mathbb{Z}[q^{1/2},q^{-1/2}]\), where \(q\) is an indeterminate. Using elementary methods the authors give a self-contained proof that the Specht module \(S^\lambda\), as defined by R. Dipper and G. James [in Proc. Lond. Math. Soc., III. Ser. 52, 20-52 (1986; Zbl 0587.20007)], is \(H\)-isomorphic to a module denoted by \(S_{w_J}\) which is explicitly defined by D. Kazhdan and G. Lusztig and is seen to afford the cell representation [of Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] associated with the right cell containing the longest element \(w_J\) in a parabolic subgroup \(W_J\) of \(W\) generated by appropriate \(J\subset S\). The authors give the association between \(J\) and \(\lambda\) explicitly.

MSC:

20C30 Representations of finite symmetric groups
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory

Software:

CHEVIE; GAP
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Full Text: DOI

References:

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