Enriched chain polytopes

Abstract

Stanley introduced a lattice polytope ${\mathcal{C}}_P$ arising from a finite poset $P$, which is called the chain polytope of $P$. The geometric structure of ${\mathcal{C}}_P$ has good relations with the combinatorial structure of $P$. In particular, the Ehrhart polynomial of ${\mathcal{C}}_P$ is given by the order polynomial of $P$. In the present paper, associated to $P$, we introduce a lattice polytope ${\mathcal{E}}_P$, which is called the enriched chain polytope of $P$, and investigate geometric and combinatorial properties of this polytope. By virtue of the algebraic technique on Gröbner bases, we see that ${\mathcal{E}}_P$ is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the $h^{\ast }$-polynomial of ${\mathcal{E}}_P$ is equal to the $h$-polynomial of a flag triangulation of a sphere. On the other hand, by showing that the Ehrhart polynomial of ${\mathcal{E}}_P$ coincides with the left enriched order polynomial of $P$, it follows from works of Stembridge and Petersen that the $h^{\ast }$-polynomial of ${\mathcal{E}}_P$ is $\gamma$-positive. Stronger, we prove that the $\gamma$-polynomial of ${\mathcal{E}}_P$ is equal to the $f$-polynomial of a flag simplicial complex.

11 pages, v2: minor revision