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

TL;DR

This paper introduces polytopes from bipartite graphs linked to root systems, showing their reflexivity, unimodular triangulations, and $ ext{γ}$-positivity, with connections to interior polynomials and real-rootedness.

Contribution

It establishes a novel connection between bipartite graphs, reflexive polytopes, and interior polynomials, revealing new combinatorial and algebraic properties.

Findings

Reflexivity of ${f B}_G$ characterizes bipartite graphs.

${f B}_G$ admits a unimodular triangulation when $G$ is bipartite.

The $h^*$-polynomial is $ ext{γ}$-positive and unimodal for bipartite graphs.

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 $h^*$-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 $h^*$-polynomials. In fact, if $G$ is bipartite, then the $h^*$-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 $h^*$-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 $h^*$-polynomial is not real-rooted but $\gamma$-positive, and coincides with the $h$-polynomial of a flag triangulation of a sphere.