Characteristic polynomial of $\overline{\mathcal{M}}_{0,n}$ and log-concavity
Jinwon Choi, Young-Hoon Kiem, Donggun Lee
TL;DR
This work defines and analyzes the characteristic polynomial S_V(m,t) attached to symmetric-function data and to graded S_n-representations arising in geometry, extending Stanley's specialization to a two-variable invariant. It demonstrates that, in geometric contexts such as products of projective spaces, GIT moduli spaces, Hessenberg varieties, and especially the moduli space \overline{\mathcal{M}}_{0,n}, these polynomials exhibit log-concavity phenomena and permit recursive computation via rooted-tree combinatorics. The authors develop a cohesive framework linking plethysm, the Stanley map, and a tree-based combinatorial model to produce explicit recursions for the characteristic polynomials of \overline{\mathcal{M}}_{0,n} and its relatives, along with asymptotic formulas and log-concavity results that generalize prior Betti-number asymptotics. The results yield new structural insights into the representation theory of symmetric groups acting on moduli spaces and provide both computational tools and conjectural directions for further log-concavity phenomena in high-dimensional geometric contexts. They also connect colorings interpretations of colorings of inputs on weighted rooted trees to the chromatic-type invariants, enriching the combinatorial understanding of these geometric invariants.
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.