A branching rule for partition complexes

TL;DR

This paper introduces a new $Y$-equivariant decomposition of the partition complex $\

Contribution

It provides a novel $Y$-equivariant decomposition of the partition complex $\

Findings

New decomposition of $\

Insights into quotient spaces of $\

Enhanced understanding of symmetric group actions

Abstract

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