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.

对于有限交换p群A和子群$\Gamma \le \operatorname {\mathrm {Aut}}(A)$，我们说对$(\Gamma ,A)$是融合可实现的，如果有有限p群$S\ge A$上的饱和融合系统${\mathcal {F}}$使得$C_S(A)=A$ , $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $作为$\operatorname {\mathrm {Aut}}(A)$的子群，和。在本文中，我们开发了工具来证明某些表示在这个意义上是不可融合实现的。例如，我们表明，对于Mathieu 群中的$p=2$或$3$和$\Gamma $来说，唯一可融合实现的${\mathbb {F}}_p\Gamma $模块（最多简单模块的扩展）是托德模块，在某些情况下是它们的对偶模块。