论文标题
融合系统中某些表示的不可销
Nonrealizability of certain representations in fusion systems
论文作者
论文摘要
对于有限的Abelian $ p $ -group $ a $和子组$γ\ le \ text {aut}（a）$，我们说，如果有饱和的融合系统$ \ nathcal {f} $，则可以实现这对$（γ，a）$是可以实现的。 $ \ textrm {aut} _ {\ Mathcal {f}}（a）=γ$作为$ \ text {aut}（a）$的子组，而$ a $在$ \ Mathcal {f} $中不正常。在本文中，我们开发了工具，以表明某些表示在这个意义上是不可实现的。例如，我们表明，$ p = 2 $或$ 3 $和$γ$是Mathieu组之一，唯一的$ \ mathbb {f}_pγ$ - 模块可融合可实现（琐碎模块的扩展）是TODD模块，在某些情况下是其Duals。
For a finite abelian $p$-group $A$ and a subgroup $Γ\le\text{Aut}(A)$, we say that the pair $(Γ,A)$ is fusion realizable if there is a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S\ge A$ such that $C_S(A)=A$, $\textrm{Aut}_{\mathcal{F}}(A)=Γ$ 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 $Γ$ one of the Mathieu groups, that the only $\mathbb{F}_pΓ$-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.
