An observation on the module structure of block algebras

Abstract

Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is invariant under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact form a semicharacteristic biset for the fusion system on S induced by G. The parameterization of such semicharacteristic bisets can then be applied to relate the module structure and defect theory of B. §0. Introduction. Let G be a finite group and S a Sylow p-subgroup of G. The left and right multiplicative actions of S on G give a partition of G by double cosets G = m i=1

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

“…Theorem 6.4, together with Proposition 6.3, recovers the main result of Gelvin [2], which asserts that, in the notation of the proposition, any S S -stable basis of OGb is F S .G/-semicharacteristic.…”

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

“…So Ω ∆(φ) = Ω Q , and similarly for Ω P . Theorem 6.4, together with Proposition 6.3, recovers the main result of Gelvin [2], which asserts that, in the notation of the proposition, any S×S-stable basis of OGb is F S (G)semicharacteristic.…”