Two Double Poset Polytopes

To every poset $P$, Stanley [Discrete Comput. Geom., 1 (1986), pp. 9--23] associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of $P$. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer [J. Combin. Theory Ser. A, 118 (2011), pp. 1322--1333] introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labeled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we c...