The Action of Young Subgroups on the Partition Complex

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.