Quadratic Gröbner bases of twinned order polytopes

Let PP and QQ be finite partially ordered sets on [d]={1,…,d}[d]={1,…,d}, and O(P)⊂RdO(P)⊂Rd and O(Q)⊂RdO(Q)⊂Rd their order polytopes. The twinned order polytope of PP and QQ is the convex polytope Δ(P,−Q)⊂RdΔ(P,−Q)⊂Rd which is the convex hull of O(P)∪(−O(Q))O(P)∪(−O(Q)). It follows that the origin of RdRd belongs to the interior of Δ(P,−Q)Δ(P,−Q) if and only if PP and QQ possess a common linear extension. It will be proved that, when the origin of RdRd belongs to the interior of Δ(P,−Q)Δ(P,−Q), the toric ideal of Δ(P,−Q)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 PP and QQ possess a common linear extension, then the twinned order polytope Δ(P,−Q)Δ(P,−Q) is a normal Gorenstein Fano polytope.