# On the canonical ideal of the Ehrhart ring of the chain polytope of a poset

## TL;DR

This paper investigates the algebraic properties of the Ehrhart ring of the chain polytope of a poset, linking its canonical ideal's properties to the combinatorial structure of the poset.

## Contribution

It characterizes when the Ehrhart ring of the chain polytope is level or anticanonical level based on poset structure and explores properties of its canonical ideal.

## Key findings

- If K[C(P)] is level, then K[O(P)] is also level.
- Symbolic and ordinary powers of the canonical ideal coincide.
- Degrees of generators of canonical and anticanonical ideals are consecutive integers.

## Abstract

Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] by a combinatorial structure of P. In particular, we show that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We exhibit examples which show the converse does not hold. Moreover, we show that the symbolic powers of the canonical ideal of K[C(P)] are identical with ordinary ones and degrees of the generators of the canonical and anticanonical ideals are consecutive integers.