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

Benjamin Homan

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

Combinatorics 2026-02-26 v1 Representation Theory

Authors: Benjamin Homan

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

Keywords

equivariant log concavity FI^\sharp-module cohomology configuration space representation stability

Cite

@article{arxiv.2602.21578,
  title  = {Equivariant log concavity and the $\operatorname{FI^\sharp}$-module structure on $H^i(\operatorname{Conf}(n,\mathbb{R}^d))$},
  author = {Benjamin Homan},
  journal= {arXiv preprint arXiv:2602.21578},
  year   = {2026}
}

Comments

11 pages. For code used in this paper, see https://github.com/bhoman-math/ELCandFISharp

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

Abstract

Keywords

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