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

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.

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