A classification of small operators using graph theory

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.