Gomi Yasushi

In this paper, we determine (tau q) over tilde (X-q(lambda)), the Gauss sums on the Iwahori-Hecke algebras of type A for irreducible characters X-q(lambda), which are q-analogues of those on the symmetric groups. We also explicitly determine the values of the corresponding trace function Psi((n))(q) = Sigma(lambda 1-n) (tau(q)) over tilde (X-q(lambda))X-q(lambda).

We shall discuss Gauss sums on finite groups and give several examples including the case of the complex reflection groups G(m, r, n), and hence finite symmetric groups, and also finite Weyl groups.

For most of the finite subgroups of SL(3, C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae [McKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme Hilb (G)(C-3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C-3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from Hilb(G)(C-3) to C-3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

In this article we study by combinatorial methods presentations of the pure braid group associated with any Coxeter group. We generalize to all types the decomposition of the pure braid group into successive semi-direct products known in the case of type A(n).

The semisimplicity of Iwahori-Hecke algebras has been studied by several authors. A. Gyoja (J. Algebra 174, 1995, 553-572) gave a necessary and sufficient condition for Iwahori-Hecke algebras to be semisimple, using the modular representation theory. The author (J. Algebra 183, 1996, 514-544) studied the semisimplicity of parabolic Hecke algebras when they have only one parameter q. In this paper we completely determine the cases when parabolic Hecke algebras are semisimple complementing our previous work applying the method of Gyoja. (C) 1998 Academic Press.

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In sectional sign 1, we discuss the structure of Hecke algebras and in sectional sign 2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness.

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups when they are commutative. In the case when Weyl groups are of classical type, they are already known in [D.1] and [D.2]. In 1, we discuss the structure of Hecke algebras and in 2, we calculate all the character tables of these commutative Hecke algebras associated with exceptional Weyl groups.