Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$

TL;DR

This paper proves a stronger form of equivariant log concavity for the cohomology of configuration spaces using representation stability, confirming the conjecture in low degrees and proposing it holds universally.

Contribution

It introduces a novel approach using $ ext{FI}^ ext{sharp}$-modules to establish equivariant log concavity for configuration space cohomology, extending known results.

Findings

Proves equivariant log concavity up to degree 19

Establishes $ ext{FI}^ ext{sharp}$-module structure on cohomology

Conjectures universal equivariant log concavity for all degrees

Abstract

Previous work has conjectured that the graded $\mathfrak{S}_n$-representations $H^\bullet(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging the theory of representation stability, we are able instead prove a stronger statement about the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ which implies the original conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the $\operatorname{FI^\sharp}$-modules $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$.