Quadratic Gröbner bases of twinned order polytopes

Abstract

Let $P$ and $Q$ be finite partially ordered sets on $[d]=\{1,\dots ,d\}$, and $\mathcal{O}(P)\subset {\mathbb{R}}^d$ and $\mathcal{O}(Q)\subset {\mathbb{R}}^d$ their order polytopes. The twinned order polytope of $P$ and $Q$ is the convex polytope $\Delta (P,-Q)\subset {\mathbb{R}}^d$ which is the convex hull of $\mathcal{O}(P)\cup (-\mathcal{O}(Q))$. It follows that the origin of ${\mathbb{R}}^d$ belongs to the interior of $\Delta (P,-Q)$ if and only if $P$ and $Q$ possess a common linear extension. It will be proved that, when the origin of ${\mathbb{R}}^d$ belongs to the interior of $\Delta (P,-Q)$, the toric ideal of $\Delta (P,-Q)$ possesses a quadratic Gröbner basis with respect to a reverse lexicographic order for which the variable corresponding to the origin is smallest. Thus in particular if $P$ and $Q$ possess a common linear extension, then the twinned order polytope $\Delta (P,-Q)$ is a normal Gorenstein Fano polytope.

6 pages

References in corpus (1)

• Centrally symmetric configurations of order polytopes

Cited by in corpus (2)

• Quadratic Gröbner bases arising from partially ordered sets
• Volume, facets and dual polytopes of twinned chain polytopes