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

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

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 -representations 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 -module structure on which implies the original conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the -modules .
Paper Structure (5 sections, 9 theorems, 14 equations)

This paper contains 5 sections, 9 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Representation Stability
  3. Stabilization of $\operatorname{FB}$-Modules
  4. FI$^\sharp$-Modules
  5. Proof

Key Result

Theorem 1.2

Given positive integers $i<j\leq k<\ell$ with $i+\ell =j+k=m$, for $m\leq 19$ we have an inclusion of $\operatorname{FI^\sharp}$-modules and

Theorems & Definitions (37)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • ...and 27 more