The action of Young subgroups on the partition complex

Gregory Z. Arone; D. Lukas B. Brantner

doi:10.1007/s10240-021-00123-7

Article

The action of Young subgroups on the partition complex

Gregory Z. Arone ¹; D. Lukas B. Brantner ¹

¹

Publications Mathématiques de l'IHÉS, Volume 133 (2021), pp. 47-156

Abstract

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 $G \subset Σ_{n}$ and prove a formula for the restriction of $| Π_{n} |$ to Young subgroups $Σ_{n_{1}} \times \dots \times Σ_{n_{k}}$ . 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 $| Π_{n} |^{⋄} \land_{Σ_{n}}^{} {(S^{ℓ})}^{\land n}$ 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” $| Π_{d} |^{⋄} \land_{Σ_{d}}^{} {(S^{ℓ})}^{\land d}$ 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.

Received: 2018-01-24
Accepted: 2021-03-11
Online First: 2021-03-25
Published online: 2021-06-07

Zbl

DOI: 10.1007/s10240-021-00123-7

Author's affiliations:

Gregory Z. Arone ¹; D. Lukas B. Brantner ¹

¹

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     title = {The action of {Young} subgroups on the partition complex},
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Gregory Z. Arone; D. Lukas B. Brantner. The action of Young subgroups on the partition complex. Publications Mathématiques de l'IHÉS, Volume 133 (2021), pp. 47-156. doi: 10.1007/s10240-021-00123-7

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