Smith Normal Form in Combinatorics

TL;DR

This paper surveys combinatorial aspects of Smith normal form, exploring algebraic properties, interpretations, and applications to graph critical groups, random matrices, symmetric functions, and hyperplane arrangements.

Contribution

It provides a comprehensive overview of Smith normal form in combinatorics, including new examples and interpretations from various mathematical contexts.

Findings

Analysis of algebraic properties of Smith normal form

Examples from symmetric functions and hyperplane arrangements

Connections to critical groups of graphs and random matrices

Abstract

This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.