Reflexive polytopes arising from bipartite graphs with $\gamma$-positivity associated to interior polynomials

Abstract

In this paper, we introduce polytopes ${\mathcal{B}}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal{B}}_G$ is reflexive if and only if $G$ is bipartite. Moreover, in the case, ${\mathcal{B}}_G$ has a regular unimodular triangulation. This implies that the $h^{\ast }$-polynomial of ${\mathcal{B}}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, namely the $\gamma$-positivity and the real-rootedness of the $h^{\ast }$-polynomials. In fact, if $G$ is bipartite, then the $h^{\ast }$-polynomial of ${\mathcal{B}}_G$ is $\gamma$-positive and its $\gamma$-polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $h^{\ast }$-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers--Stanley conjecture, we construct a bipartite graph $G$ whose $h^{\ast }$-polynomial is not real-rooted but $\gamma$-positive, and coincides with the $h$-polynomial of a flag triangulation of a sphere.

20 pages, 5 figures, many explanations are added, References are added and updated