A family of matrix-tree multijections

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.

arXiv commentSeveral edits from the previous version including a new title (the previous title was "A Combinatorial Mapping for the Higher-Dimensional Matrix-Tree Theorem"). There are also many added references to the author's dissertation

References in corpus (5)

• Chip-Firing and Rotor-Routing on Directed Graphs
• Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory
• Orientable arithmetic matroids
• A family of matrix-tree multijections
• Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

Cited by in corpus (2)

• A family of matrix-tree multijections
• Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs