A classification of small operators using graph theory

TL;DR

This paper classifies small non-negative integer matrices with operator norm less than 2, linking them to bipartite graphs and Coxeter graphs, and explores their relation to quadratic forms, reflection groups, and Lie algebras.

Contribution

It provides a classification of small matrices via graph theory, connecting spectral properties to algebraic structures like Coxeter graphs and Lie algebras.

Findings

Matrices with norm < 2 correspond to Coxeter graphs.

Classification aligns with ADE-type graphs.

Establishes links between spectral graph theory and algebraic structures.

Abstract

Given a real $n \times m$ matrix $B$, its operator norm can be defined as $$|B|=\max_{|v|=1}|Bv|.$$ We consider a matrix "small" if it has non-negative integer entries and its operator norm is less than $2$. These matrices correspond to bipartite graphs with spectral radius less than $2$, which can be classified as disjoint unions of Coxeter graphs. This gives a direct route to an $ADE$-classification result in terms of very basic mathematical objects. Our goal here is to see these results as part of a general program of classification of small objects, relating quadratic forms, reflection groups, root systems, and Lie algebras.