# Enriched order polytopes and Enriched Hibi rings

## TL;DR

This paper introduces enriched order polytopes and enriched Hibi rings, revealing their reflexivity, Ehrhart polynomial equivalence, and algebraic properties, thus extending Stanley's lattice polytope theory to enriched structures.

## Contribution

It defines enriched order polytopes and Hibi rings, proves their key properties, and establishes connections with enriched P-partitions and algebraic structures.

## Key findings

- Enriched order polytopes are reflexive and have Ehrhart polynomials matching enriched chain polytopes.
- Enriched Hibi rings are shown to be normal, Gorenstein, and Koszul.
- A bijection between lattice points of enriched polytopes is established.

## Abstract

Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope ${\mathcal O}_P$ and the chain polytope ${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ are equal to the order polynomial of $P$ that counts the $P$-partitions. In this paper, we introduce the enriched order polytope of a poset $P$ and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of $P$ and the left enriched order polynomial of $P$ that counts the left enriched $P$-partitions, by using the theory of Gr\"{o}bner bases. The toric rings of enriched order polytopes are called enriched Hibi rings. It turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The above result implies the existence of a bijection between the lattice points in the dilations of ${\mathcal O}^{(e)}_P$ and ${\mathcal C}^{(e)}_P$. Towards such a bijection, we give the facet representations of enriched order and chain polytopes.