Two double poset polytopes

TL;DR

This paper introduces double order and chain polytopes for double posets, exploring their geometric properties, combinatorial structures, and algebraic aspects, thereby extending Stanley's classical polytopes and connecting to optimization and algebraic geometry.

Contribution

It defines and analyzes double poset polytopes, describing their facial structures, Ehrhart polynomials, volumes, and algebraic properties, and establishes connections to anti-blocking polytopes and Groebner bases.

Findings

Double poset polytopes capture interactions between two partial orders.

Canonical subdivisions facilitate Ehrhart polynomial and volume computations.

Quadratic Groebner bases describe associated semigroup rings.

Abstract

To every poset P, Stanley (1986) 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 (2011) introduced 'double posets', that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled 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 characterize 2-level polytopes among our double poset polytopes. Fulkerson's 'anti-blocking' polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets. We determine the geometry of Minkowski- and Cayley sums of anti-blocking polytopes. In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-)polynomials and volumes. This also yields canonical triangulations of double poset polytopes. Finally, we investigate the affine semigroup rings associated to double poset polytopes. We show that they have quadratic Groebner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part.