The action of Young subgroups on the partition complex

Abstract We study the restrictions, strict fixed points, and quotients of partition complex $|\Pi_{n}|$ | Π n , which is $\Sigma_{n}$ Σ -space attached to poset proper nontrivial partitions set $\{1,\ldots,n\}$ { 1 , … } . express space points $|\Pi_{n}|^{G}$ G in terms subgroup posets for general $G\subset \Sigma_{n}$ ⊂ prove a formula restriction Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ × ⋯ k Both results follow by applying method, proven with discrete Morse theory, producing equivariant branching rules on lattices group actions. uncover surprising links between commutative monoid spaces, cotangent fibre derived algebraic geometry. These connections allow us construct cofibre sequence relating various $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ ⋄ ∧ ( S ℓ ) give combinatorial proof splitting Combining all our results, we decompose “atoms” $|\Pi_{d}|^{\diamond} }_{\Sigma_{d}}^{}} d}$ d $\ell$ odd compute their homology. thereby also generalise Goerss’ computation André-Quillen homology trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 {F}_{p}$ p $p$ an prime.