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 \(\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}\). 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 \(|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}\) 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” \(|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}\) for \(\ell\) odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from \(\mathbf {F}_{2}\) to \(\mathbf {F}_{p}\) for \(p\) an odd prime.
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Arone, G.Z., Brantner, D.L.B. The action of Young subgroups on the partition complex. Publ.math.IHES 133, 47–156 (2021). https://doi.org/10.1007/s10240-021-00123-7
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DOI: https://doi.org/10.1007/s10240-021-00123-7