2017/10/21 by Bisson, Terrence, Lopez, Jonathan
#05C50 #15A60 #15B36 #17B20 #17B22 #20F55 #Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO)
paper · doi:10.48550/arxiv.1710.07809
Given a real n × 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.