The action of Young subgroups on the partition complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex | Π n | , which is the Σ n -space attached to the poset of proper nontrivial partitions of the set { 1 , … , n } . We express the space of fixed points | Π n | G in terms of subgroup posets for general G ⊂ Σ n and prove a formula for the restriction of | Π n | to Young subgroups Σ n 1 × ⋯ × Σ 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 | Π 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 | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of | Π n | in terms of “atoms” | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from F 2 to F p for p an odd prime.