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

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 $H^i(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$.