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Representations of Hecke algebras at roots of unity. (English) Zbl 0931.20006

Séminaire Bourbaki. Volume 1997/98. Exposés 835–849. Paris: Société Mathématique de France, Astérisque. 252, 33-55, Exp. No. 836 (1998).
This is an expository paper on recent developments in the representation theory of Hecke algebras associated with \(p\)-adic Chevalley groups \(G\), especially on the \(\ell\)-decomposition matrices of such algebras, where \(\ell\neq p\). The Hecke algebra is defined in general setting in terms of the Coxeter system \((W,R)\) of \(G\) and the paper explains many open problems and conjectures in this area such as a conjecture formulated by G. James for the case when \(W=S_n\), the symmetric group, and the Lascoux-Leclerc-Thibon conjecture.
For the entire collection see [Zbl 0911.00019].

MSC:

20C08 Hecke algebras and their representations
20C20 Modular representations and characters
20G05 Representation theory for linear algebraic groups
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
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