Order-chain polytopes

Abstract

Given two families X and Y of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate a new class of polytopes is to take the intersection P = P1 ∩ P2, where P1 ∈ X, P2 ∈ Y . Two basic questions then arise: 1) when P is integral and 2) whether P inherits the “old type” from P1, P2 or has a “new type”, that is, whether P is unimodularly equivalent to a polytope in X∪Y or not. In this paper, we focus on the families of order polytopes and chain polytopes. Following the above framework, we create a new class of polytopes which are named order-chain polytopes. When studying their volumes, we discover a natural relation with Ehrenborg and Mahajan’s results on maximizing descent statistics.