NONREALIZABILITY OF CERTAIN REPRESENTATIONS IN FUSION SYSTEMS

Abstract For a finite abelian p -group A and a subgroup $\Gamma \le \operatorname {\mathrm {Aut}}(A)$ , we say that the pair $(\Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${\mathcal {F}}$ over a finite p -group $S\ge A$ such that $C_S(A)=A$ , $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $ as subgroups of $\operatorname {\mathrm {Aut}}(A)$ , and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma $ one of the Mathieu groups, that the only ${\mathbb {F}}_p\Gamma $ -modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.