Abstract
For a natural class of r×n integer matrices, we construct a non-convex polytope which periodically tiles 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.
Highlights
Given a connected graph G, the sandpile group S(G) is a finite abelian group related to a discrete dynamical system
In the author’s dissertation, he shows that any graph, regular matroid, cell complex with a torsion-free spanning forest, or orientable arithmetic matroid with a multiplicity one basis is associated with a standard representative matrix [20]
In 2017, Backman, Baker, and Yuen define a family of geometric bijections between S(D) and B(D) for the regular matroid case [2, 24]
Summary
Given a connected graph G, the sandpile group S(G) is a finite abelian group related to a discrete dynamical system. In the author’s dissertation, he shows that any graph, regular matroid, cell complex with a torsion-free spanning forest, or orientable arithmetic matroid with a multiplicity one basis is associated with a standard representative matrix [20]. In this context, we get the following theorem, which is a reframing of Theorem 8.1 from [11]. In 2017 (published in 2019), Backman, Baker, and Yuen define a family of geometric bijections between S(D) and B(D) for the regular matroid case [2, 24] Their construction does not generalize to the case where not all bases have multiplicity 1.
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