论文标题
矩阵树多注射器的家族
A family of matrix-tree multijections
论文作者
论文摘要
对于天然类别的$ r \ times n $整数矩阵，我们构建了一个非凸polytope，它会定期瓷砖$ \ mathbb r^n $。从这个瓷砖中，我们提供了一个从广义的沙珀组到一组广义的跨越树的几何有意义的地图，为几个高维矩阵Tree定理提供了多主体证明。特别是，这些多注射器可以应用于图形，常规矩形，具有无扭转的跨越森林的细胞复合物以及具有多种基础的可代表算术矩阵。这概括了Backman，Baker和Yuen进行的两次培养，并扩展了Duval，Klivans和Martin的工作。
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.
