Abstract. We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.

Abstract. We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.

No abstract

This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two p-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for p-blocks of alternating groups (where the blocks must also have the same sign when p is odd), of double covers of alternating and symmetric groups (for p odd, and where we obtain crossover isometries when the blocks have opposite signs), of complex reflection groups G(d, 1, n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks, in a way which should be of independent interest.

No abstract

This note is concerned with the McKay conjecture in the representation theory of finite groups. Recently, Isaacs-Malle-Navarro have shown that, in order to prove this conjecture in general, it is sufficient to establish certain properties of all finite simple groups. In this note, we develop some new methods for dealing with these properties for finite simple groups of Lie type in the defining characteristic case. We apply these methods to show that the Suzuki and Ree groups, G 2 (q), F 4 (q) and E 8 (q) have the required properties.

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups PΩ 2m+1 (p n ) and PSp 2m (p n ) are "good" for p > 2 and the simple groups E 7 (p n ) are "good" for p > 3 in the sense of Isaacs, Malle and Navarro. We also describe the action of the diagonal and field automorphisms on the semisimple and the regular characters.

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field 𝔽q of characteristic p > 0 with corresponding Frobenius map F. We prove that, if p is a good prime for G and if the group of F‐coinvariants of the component group of the centre of G has prime order, then the relative McKay conjecture holds for GF at the prime p. In particular, this conjecture is true for GF in the defining characteristic for a simple simply connected group G of type Bn, Cn, E6 or E7. Our main tools are the theory of Gelfand–Graev characters for connected reductive groups with disconnected centre developed by Digne, Lehrer and Michel and the theory of cuspidal Levi subgroups. We also explicitly compute the number of semisimple classes of GF for any simple algebraic group G.