On the action of Young subgroups on the complex of partitions

The action of the symmetric group $S_n$ on the complex of partitions $P_n$ has received considerable attention. We study the restriction of this action to a Young subgroup of $S_n$. Let $G$ be a Young subgroup of $S_n$. Our main result is a $G$-equivariant decomposition of $P_n$. As an application, we describe the fixed point space $(P_n)^H$ in terms of subgroup posets, for every subgroup $H$ of $S_n$. For another application we obtain new information about the orbit space $(P_n)_G$, where $G$ is a Young subgroup.