A branching rule for partition complexes

Let $S_n$ be the symmetric group, and let $Y$ be a Young subgroup of $S_n$. Let ${\mathrm{\Pi }}_n$ be the complex of partitions of $\{1,\dots ,n\}$. Our main result is a $Y$-equivariant decomposition of ${\mathrm{\Pi }}_n$. As an application, we obtain new information about the quotient space of $\mathrm{\mid }{\mathrm{\Pi }}_n\mathrm{\mid }$ by a Young subgroup.