Quadratic Gröbner bases of twinned order polytopes

Abstract

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