Conjectural invariance with respect to the fusion system of an almost-source algebra

Abstract

We show that, given an almost-source algebra A of a p-block of a finite group G, then the unit group of A contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when G is p-solvable, those two equivalent conditions hold for some almost-source algebra of the given p-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for A, then any stable basis for A is semicharacteristic for the fusion system.

Inverse images of block varieties

“…It is not known whether every block B with defect group P has at least some almost source idempotent i ∈ B P such that the almost source algebra iBi has a P -P -stable basis consisting of invertible elements. See [3] for equivalent reformulations of this problem, as well as a number of cases in which this is true. The following technical observation is a special case of Puig's characterisation of fusion in source algebras in [22].…”

“…) . Barker and Gelvin conjectured in [3], that every block with a defect group P should indeed have an almost source algebra with a P -P -stable basis consisting of invertible elements. If F = N F (P ) and i a source idempotent, then it is easy to show that iM is F -stable for any finitely generated B-module M .…”