Characteristic polynomial of $\overline{\mathcal{M}}_{0,n}$ and log-concavity

TL;DR

This paper introduces a characteristic polynomial for symmetric group representations, explores its properties in geometric contexts, and demonstrates asymptotic log-concavity in the cohomology of moduli spaces of pointed rational curves.

Contribution

It generalizes the characteristic polynomial concept to symmetric functions and representations, and proves asymptotic log-concavity for specific geometric cases.

Findings

Coefficients of the characteristic polynomial tend to be log-concave when arising from geometry.

Asymptotic formulas for the characteristic polynomial of moduli space cohomology are established.

Asymptotic log-concavity is proven for the cohomology of the moduli space of pointed rational curves.

Abstract

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric function. When the representation arises from geometry, the coefficients of its characteristic polynomial tend to form a log-concave sequence. To illustrate, we investigate explicit examples, including the $n$-fold products of the projective spaces, the GIT moduli spaces of points on $\mathbb{P}^1$ and Hessenberg varieties. Our main focus lies on the cohomology of the moduli space of pointed rational curves, for which we prove asymptotic formulas of its characteristic polynomial and establish asymptotic log-concavity.