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Pré-Publication, Document De Travail Année : 2022

Nonrealizability of certain representations in fusion systems

Bob Oliver
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  • PersonId : 852620

Résumé

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.

Dates et versions

hal-03840893 , version 1 (06-11-2022)

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Bob Oliver. Nonrealizability of certain representations in fusion systems. 2022. ⟨hal-03840893⟩
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