On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope

TL;DR

This paper explores the structure and volume of an alternating sign matrix polytope analogue, revealing connections to order and flow polytopes, and compares different triangulations with explicit bijections.

Contribution

It introduces the ASM-CRY polytope, establishes its combinatorial properties, and relates various triangulations of order and flow polytopes, including explicit bijections.

Findings

ASM-CRY polytope has Catalan many vertices.

Volume equals the number of standard Young tableaux of staircase shape.

Triangulations of flow polytopes are related through explicit bijections.

Abstract

In this paper we study an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaux of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley's triangulation of order polytopes, the Postnikov-Stanley triangulation of flow polytopes and the Danilov-Karzanov-Koshevoy triangulation of flow polytopes. We show that when a graph $G$ is a planar graph, in which case the flow polytope $F_G$ is also an order polytope, Stanley's triangulation of this order polytope is one of the Danilov-Karzanov-Koshevoy triangulations of $F_G$. Moreover, for a general graph $G$ we show that the set of Danilov-Karzanov-Koshevoy triangulations of $F_G$ is a subset of the set of Postnikov-Stanley triangulations of $F_G$. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.