The Action of Young Subgroups on the Partition Complex

TL;DR

This paper investigates the action of Young subgroups on the partition complex, providing formulas for fixed points and quotients, and linking these to algebraic and geometric structures, including derived algebraic geometry and homology computations.

Contribution

It introduces a general method using discrete Morse theory to analyze subgroup actions on the partition complex and connects these to algebraic and geometric frameworks.

Findings

Derived formulas for fixed points of subgroup actions.

Constructed cofibre sequences relating strict quotients.

Generalized homology computations for partition complexes.

Abstract

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_n|$, which is the $\Sigma_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|\Pi_n|^G$ in terms of subgroup posets for general $G\subset \Sigma_n$ and prove a formula for the restriction of $|\Pi_n|$ to Young subgroups $\Sigma_{n_1}\times \dots\times \Sigma_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover surprising links between strict Young quotients of $|\Pi_n|$, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_n|^\diamond\wedge_{\Sigma_n} (S^\ell)^{\wedge n}$ and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of $|\Pi_n|$ in terms of "atoms" $|\Pi_d|^\diamond\wedge_{\Sigma_d} (S^\ell)^{\wedge d}$ for $\ell$ odd and compute their homology. We thereby also generalise Goerss' computation of the algebraic Andr\'e-Quillen homology of trivial square-zero extensions from $\mathbb{F}_2$ to $\mathbb{F}_p$ for $p$ an odd prime.