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

Hidefumi Ohsugi, Akiyoshi Tsuchiya

Published 2018-10-29Version 1

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 a reflexive polytope with a regular unimodular triangulation if and only if $G$ is bipartite. This implies that the $h^*$-polynomial of ${\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, 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 Tutte polynomial of a hypergraph). Moreover, the $h^*$-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted.