Nonrealizability of certain representations in fusion systems

TL;DR

This paper investigates which group representations can be realized within fusion systems, demonstrating that certain representations, such as those of Mathieu groups, are not fusion realizable except for specific modules like Todd modules.

Contribution

The authors develop new tools to prove the nonrealizability of certain representations in fusion systems, especially for Mathieu groups and specific modules.

Findings

Certain Mathieu group representations are not fusion realizable.

Only Todd modules and their duals are fusion realizable for these groups.

The paper provides criteria to determine nonrealizability of representations.

Abstract

For a finite abelian $p$-group $A$ and a subgroup $\Gamma\le\text{Aut}(A)$, 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 $C_S(A)=A$, $\textrm{Aut}_{\mathcal{F}}(A)=\Gamma$ as subgroups of $\text{Aut}(A)$, and $A$ is not normal in $\mathcal{F}$. 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.