Meinolf Geck

CHEVIE — A system for computing and processing generic character tables

CHEVIE is a computer algebra package which collects data and programs for the representation theory of finite groups of Lie type and associated structures. We explain the theoretical and conceptual background of the various parts of CHEVIE and we show the usage of the system by means of explicit examples. More precisely, we have sections on Weyl groups and Iwahori-Hecke algebras, generic character tables of series of finite groups of Lie type, and cyclotomic algebras.

Representations of Hecke algebras at roots of unity

Generic Iwahori-Hecke algebras.- Kazhdan-Lusztig cells and cellular bases.- Specialisations and decomposition maps.- Hecke algebras and finite groups of Lie type.- Representation theory of Ariki-Koike algebras.- Canonical bases in affine type A and Arikis theorem.- Decomposition numbers for exceptional types.

Centers and simple modules for Iwahori-Hecke algebras

The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type A n has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are “generic” features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the “genericity” of the number of simple modules of Iwahori-Hecke algebras.

Hecke algebras of finite type are cellular

Let

Weights of Markov traces and generic degrees

\mathcal{H}

Irreducible brauer characters of the 3-dimensional special unitary groups in non-defining characteristic ∗

be the one-parameter Hecke algebra associated to a finite Weyl group W, defined over a ground ring in which “bad” primes for W are invertible. Using deep properties of the Kazhdan–Lusztig basis of

Basic sets of Brauer characters of finite groups of Lie type, III

\mathcal{H}

ON THE INDUCTION OF KAZHDAN¿LUSZTIG CELLS

and Lusztig’s a-function, we show that

Kazhdan–Lusztig Cells and the Murphy Basis

\mathcal{H}

On the existence of a unipotent support for the irreducible characters of a finite group of Lie type

has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of “Specht modules” for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types An and Bn.