Equivariant log concavity and representation stability

TL;DR

This paper explores equivariant log concavity in algebraic structures related to matroids and hyperplane arrangements, proposing conjectures and using representation stability for proofs in specific cases.

Contribution

It introduces equivariant log concavity conjectures for various algebraic structures and applies representation stability to prove them in low degrees for Coxeter arrangements.

Findings

Conjectures on equivariant log concavity for Orlik--Solomon, Cordovil, and Orlik--Terao algebras.

Computer-assisted proofs for low-degree cases in Coxeter arrangements.

Extension of representation stability techniques to algebraic log concavity problems.

Abstract

We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.