An observation on the module structure of block algebras
Matthew Gelvin

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.
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