Equivariant Log Concavity and the FI ♯ -Module Structure on H i (Conf( n ,ℝ d ))

Gedeon, Proudfoot, and Young first conjectured that the graded Sn -representations H∙(Conf(n,Rd);Q) are strongly equivariantly log concave for even d. Later work of Matherne, Miyata, Proudfoot, and Ramos conjectured strong equivariant log concavity holds for all values of d and proved this conjecture up to degree 14. By leveraging the theory of representation stability, we are able to prove a stronger statement about the FI♯ -module structure on Hi(Conf(n,Rd);Q) which implies the latter conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the FI♯ -modules Hi(Conf(n,Rd);Q).