Smith Normal Forms of Graphical Hermite Simplices
Benjamin Braun, Antwon Park
TL;DR
The paper studies the Smith normal form of vertex-matricies for graphical Hermite simplices, a class of lattice simplices formed by small lattice perturbations of rectangular simplices via a DAG on $[n]$, encoded as $A_{\vec{d},G}=\mathrm{diag}(d_1,\dots,d_n)+\sum_{(i,j)\in E(G)}E_{i,j}$. It links SNF to the cokernel of the fundamental parallelepiped, deriving both general properties and path-based determinant formulas that connect graph structure to invariant factors. It proves sufficient conditions for a cyclic cokernel (single non-unit invariant factor) under pairwise coprime diagonals, path graphs, or constant diagonal with a length-$n$ path, and provides a bound $\alpha_n\le m^h$ when the diagonal is constant $m$ and the longest path has length $h$, along with explicit SNF descriptions in bipartite cases. In the prime-diagonal setting, the SNF entries are restricted to $1,p,p^2$, with corollaries linking divisor structure to the rank of the adjacency-related matrix modulo $p$, and a conjecture about the distribution of $p$-adic factors based on $\mathbb{Z}/p\mathbb{Z}$-rank.
Abstract
We introduce the family of graphical Hermite simplices and study the Smith normal forms of their matrices of vertex vectors, which is equivalent to studying the group structure of the cokernels for these matrices. Our motivation is to study the behavior of lattice simplices subject to small lattice perturbations of their vertices. In this case, a graphical Hermite simplex is a perturbation of a rectangular simplex, i.e., a simplex defined by a diagonal matrix and the origin, with the perturbation controlled by the structure of a directed graph. We first establish sufficient conditions on the graphs and diagonal entries of these matrices that imply having a single non-unit invariant factor, i.e., a cyclic cokernel. We then obtain bounds on the invariant factors of the defining matrices related to lengths of paths in the corresponding directed graph.