A family of matrix-tree multijections

TL;DR

This paper introduces a family of multijections linking generalized sandpile groups to spanning trees across various combinatorial structures, providing new bijective proofs for higher-dimensional matrix-tree theorems.

Contribution

It constructs a novel non-convex polytope tiling and develops multijections that extend existing bijections to broader combinatorial contexts.

Findings

Provides multijections for graphs, matroids, and cell complexes.

Generalizes previous bijections to higher dimensions.

Offers new proofs for matrix-tree theorems.

Abstract

For a natural class of $r \times n$ integer matrices, we construct a non-convex polytope which periodically tiles $\mathbb R^n$. From this tiling, we provide a family of geometrically meaningful maps from a generalized sandpile group to a set of generalized spanning trees which give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be applied to graphs, regular matroids, cell complexes with a torsion-free spanning forest, and representable arithmetic matroids with a multiplicity one basis. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.