Local-global conjectures in the representation theory of finite groups

Abstract

Abstract. We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.

Jordan decomposition for weights and the blockwise Alperin weight conjecture

“…Is there an analogue of Jordan decomposition for weights of finite groups of Lie type? Malle proposed this problem in Problem 4.9 of [47]. Following Kessar-Malle [39, p. 101], we hope for a Bonnafé-Rouquier type reduction (cf.…”

The inductive blockwise Alperin weight condition for type $\mathsf C$ and the prime $2$

“…Is there an analogue of Jordan decomposition for weights of finite groups of Lie type? This question is posed in Problem 4.9 of [29]. We give a positive answer for 2-weights of G = Sp 2n (q).…”

On the inductive blockwise Alperin weight condition for classical groups

“…In this paper, the assumption that ℓ is linear is always to ensure the unitriangular shape of decomposition matrices, which is due to Gruber and Hiss [27]. It is an open problem to show that decomposition matrices of finite groups of Lie type in non-defining characteristic have unitriangular shape (see for example [43,Problem 4.8]). If this is true, then the assumption that ℓ is linear can be removed from the main results of this paper.…”