Anchors of irreducible characters

Given a prime number p, every irreducible character χ of a finite group G determines a unique conjugacy class of p-subgroups of G which we will call the anchors of χ. This invariant has been considered by Barker in the context of finite p-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other p-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J.A. Green and vertices in the sense of G. Navarro

Descent of Equivalences and Character Bijections

Categorical equivalences between block algebras of finite groups—such as Morita and derived equivalences—are well known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non-splitting fields. This article presents various results on the theme of descent to appropriate subfields and subrings. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centres in non-split situations and explain connections with Navarro’s generalisation of the Alperin–McKay conjecture. We show that Rouquier’s splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source

The structure of blocks with a Klein four defect group

We prove Erdmann’s conjecture [16] stating that every block with a Klein four defect group has a simple module with trivial source, and deduce from this that Puig’s finiteness conjecture holds for source algebras of blocks with a Klein four defect group. The proof uses the classification of finite simple groups

On a question of Külshammer for representations of finite groups in reductive groups

Acknowledgments. The authors acknowledge the financial support of EPSRC Grant EP/L005328/1, Marsden Grants UOC1009 and UOA1021, and the DFG-priority programme SPP1388 “Representation Theory”. We are grateful to the referee for helpful suggestions.Peer reviewedPostprin