Enriched chain polytopes

Abstract

Stanley introduced a lattice polytope \(\mathscr{C}_P\) arising from a finite poset P, which is called the chain polytope of P. The geometric structure of \(\mathscr{C}_P\) has good relations with the combinatorial structure of P. In particular, the Ehrhart polynomial of \(\mathscr{C}_P\) is given by the order polynomial of P. In the present paper, associated to P, we introduce a lattice polytope ℰP, 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 ℰP is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the h*-polynomial of ℰP is equal to the h-polynomial of a flag triangulation of a sphere. On the other hand, by showing that the Ehrhart polynomial of ℰP coincides with the left enriched order polynomial of P, it follows from works of Stembridge and Petersen that the h*-polynomial of ℰP is γ-positive. Stronger, we prove that the γ-polynomial of ℰP is equal to the f-polynomial of a flag simplicial complex.