Birkhoff polytopes of different type and the orthant-lattice property

TL;DR

This paper extends the study of Birkhoff polytopes to type-B Coxeter groups, proving reflexivity, Gorenstein property, and triangulation properties, and introduces the orthant-lattice property to generalize these results.

Contribution

It introduces the orthant-lattice property and applies it to establish triangulation and Gorenstein properties for type-B Birkhoff polytopes and related reflexive polytopes.

Findings

Type-B Birkhoff polytope is reflexive and Gorenstein.

The polytope and its dual have regular, unimodular triangulations.

The orthant-lattice property enables general results for reflexive polytopes.

Abstract

The Birkhoff polytope, defined to be the convex hull of $n\times n$ permutation matrices, is a well studied polytope in the context of the Ehrhart theory. This polytope is known to have many desirable properties, such as the Gorenstein property and existence of regular, unimodular triangulations. In this paper, we study analogues of the Birkhoff polytope for finite irreducible Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope $\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it and its dual polytope are reflexive, and hence Gorenstein, and also possess regular, unimodular triangulations. Noting that our triangulation proofs do not rely on the combinatorial structure of $\mathcal{BB}(n)$, we define the notion of an orthant-lattice property polytope and use this to prove more general results for the existence of regular, unimodular triangulations and unimodular covers for a significant family of reflexive polytopes. We conclude by remarking on some connections to Gale-duality, Birkhoff polytopes of other types, and possible applications of orthant-lattice property.