Matching fields and lattice points of simplices

TL;DR

This paper establishes a bijection between degree vectors and lattice points of simplices via linkage matching fields, resolving open questions and providing explicit constructions and characterizations related to triangulations, matroids, and flip graphs.

Contribution

It introduces a novel bijection linking lattice points and degree vectors in linkage matching fields, solving longstanding open problems and offering new structural insights.

Findings

Bijection between degree vectors and lattice points of simplices

Explicit construction associating bipartite graphs to lattice points

Characterization of the flip graph via prodsimplicial flag complex

Abstract

We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.