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 $a_i\in {\mathbb{R}}_{>0}$, $a_1{x}_1+\cdots +a_n{x}_n\in R_M^1$ satisfies the strong Lefschetz property (moreover, Hodge--Riemann bilinear relation) at degree one of the Artinian Gorenstein algebra $R_M^{\ast }$ associated to a graphic matroid $M$, which is defined by Maeno and Numata for all matroids.

21 pages, comments are welcome. v3: add a detailed explanation of the idea of the proof of main theorem in Key Observation 4.8. Also, add Remark 5.8 and some minor changes

Cited by in corpus (3)

• On the determinant of multiplication map of a monomial complete intersection ring
• The eigenvalues of the Hessian matrices of the generating functions for trees with $k$ components
• On Artinian Gorenstein algebras associated to the face posets of regular polyhedra