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.

Revised:

Accepted:

Published online:

Keywords: sandpile group, multijection, arithmetic matroid

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@article{ALCO_2021__4_5_795_0, author = {McDonough, Alex}, title = {A family of matrix-tree multijections}, journal = {Algebraic Combinatorics}, pages = {795--822}, publisher = {MathOA foundation}, volume = {4}, number = {5}, year = {2021}, doi = {10.5802/alco.181}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.181/} }

McDonough, Alex. A family of matrix-tree multijections. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 795-822. doi : 10.5802/alco.181. http://archive.numdam.org/articles/10.5802/alco.181/

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