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

TL;DR

This paper investigates the invariance properties of almost-source algebras of p-blocks in finite groups, linking the structure of their unit groups to fusion system invariance and establishing conditions for p-solvable groups.

Contribution

It establishes a connection between the unit group basis stabilization in almost-source algebras and fusion system invariance, providing new criteria for p-solvable groups.

Findings

Unit group basis stabilization is equivalent to fusion system invariance.

For p-solvable groups, these conditions hold for some almost-source algebra.

Stable bases are semicharacteristic for the fusion system under certain conditions.

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.