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.