$h^*$-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs

Abstract

Conditions are given on a lattice polytope $P$ of dimension $m$ or its associated affine semigroup ring which imply inequalities for the $h^*$-vector $(h^*_0, h^*_1,\dots,h^*_m)$ of $P$ of the form $h^*_i \ge h^*_{d-i}$ for $1 \le i \le \lfloor d / 2 \rfloor$ and $h^*_{\lfloor d / 2 \rfloor} \ge h^*_{\lfloor d / 2 \rfloor + 1} \ge \cdots \ge h^*_d$, where $h^*_i = 0$ for $d < i \le m$. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included.