# Enriched chain polytopes

## TL;DR

This paper introduces the enriched chain polytope associated with a poset, revealing its reflexive structure, unimodular triangulation, and connections to enriched order polynomials, advancing the understanding of polyhedral and combinatorial properties.

## Contribution

It defines the enriched chain polytope, proves its reflexivity and triangulation properties, and links its Ehrhart polynomial to enriched order polynomials, providing new insights into its combinatorial structure.

## Key findings

- $ ext{E}_{P}$ is a reflexive polytope with a flag unimodular triangulation.
- The $h^*$-polynomial of $ ext{E}_{P}$ equals the $h$-polynomial of a flag triangulation of a sphere.
- The Ehrhart polynomial of $ ext{E}_{P}$ coincides with the left enriched order polynomial of $P$.

## 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 Gr\"{o}bner bases, we see that $\mathcal{E}_P$ is a reflexive polytope with a flag regular unimodular triangulation. Moreover, the $h^*$-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 $h^*$-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.