arXiv subjects
Publications and source records attributed to Olivier Brunat.
In this note, we construct a 2-basic set of the alternating group \A_n. To do this, we construct a 2-basic set of the symmetric group \sym_n with an additional property, such that its restriction to \A_n is a 2-basic set. We adapt here a method developed in \cite{BrGr} for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Külshammer, Olsson and Robinson in \cite{KOR}.
We denote by A the ring of Laurent polynomials in the indeterminate v and by K its field of fractions. In this paper, we are interested in representation theory of the "generic" q-Schur algebra S_q(n,r) over A. We will associate to every non-degenerate symmetrising trace form τon KS_q(n,r) a subalgebra J_τ of KS_q(n,r) which is isomorphic to the "asymptotic" algebra \J(n,r)_A defined by J. Du. As a consequence, we give a new criterion for James' conjecture.
This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric group $\sym_n$ has a $p$-basic set with some additional properties, allowing us to deduce a $p$-basic set for $\A_n$. Our main tool is the generalized perfect isometries introduced by Külshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition number of $\A_n$.
This note is concerned with the unipotent characters of the Ree groups of type G_2. We determine the roots of unity associated by Lusztig and Digne-Michel to unipotent characters of ^2G_2(3^{2n+1}) and we prove that the Fourier matrix of ^2G_2(3^{2n+1}) defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G_2.
Lusztig conjectured that the almost characters of a finite reductive group are up to a scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group. We propose in this paper to verify this conjecture for the uniform almost characters of the Suzuki and the Ree groups and to show that a conjecture of the same type holds for the disconnected case of type B_2. We then state a conjecture on the possible values of the uniform almost characters on unipotent elements in the disconnected case of type F_4.
The main aim of this paper is to associate to every cuspidal unipotent character of the Suzuki group its root of unity and to give a possible defintion of the Fourier matrix associated to the family of cuspidal unipotent characters of this group. We compute to this end the Shintani descents of Suzuki groups and use results of Digne and Michel.