Quadratic Gr\"obner bases arising from partially ordered sets

TL;DR

This paper introduces a new convex polytope derived from order and chain polytopes of partially ordered sets, demonstrating its normality and Gorenstein Fano properties using toric ideal theory.

Contribution

It defines the polytope ((P), -(Q)) and proves its key geometric properties, linking combinatorics and algebraic geometry.

Findings

The polytope ((P), -(Q)) is normal.

It is Gorenstein and Fano.

The properties are shown via reverse lexicographic squarefree initial ideals.

Abstract

The order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ which is the convex hull of $\mathcal{O}(P) \cup (-\mathcal{C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.