An observation on the module structure of block algebras

TL;DR

This paper explores how the p-fusion in a finite group G influences the module structure of a p-block B, revealing that invariant bases under Sylow p-subgroup actions form semicharacteristic bisets, linking module structure to fusion systems.

Contribution

It introduces a novel connection between invariant bases of block algebras and semicharacteristic bisets, advancing understanding of module structures in relation to fusion systems.

Findings

Invariant bases form semicharacteristic bisets

Fusion constrains module structure of block B

Parameterization relates module structure to defect theory

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.