Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property

Abstract

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial {of a complete graph} can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independently interesting in its own right. Furthermore, we prove that for any $a_i\in\mathbb{R}_{>0}$, $a_1x_1+\cdots+a_nx_n\in R^1_{M}$ satisfies the strong Lefschetz property (moreover, Hodge--Riemann bilinear relation) at degree one of the Artinian Gorenstein algebra $R^*_M$ associated to a graphic matroid $M$, which is defined by Maeno and Numata for all matroids.