Quiver varieties and root multiplicities in rank 3

TL;DR

This paper extends the use of quiver varieties to rank 3 symmetric Kac-Moody algebras, providing combinatorial upper bounds on imaginary root space dimensions and an explicit method for bipartite diagrams.

Contribution

It introduces a new combinatorial approach for rank 3 algebras and describes an explicit method for bipartite Dynkin diagrams, building on previous rank 2 work.

Findings

Derived tight upper bounds for imaginary root multiplicities in rank 3

Provided an explicit combinatorial extraction method for bipartite diagrams

Computational evidence suggests bounds are close to actual values

Abstract

Building on our previous work in rank two, we use quiver varieties to give a combinatorial upper bound on dimensions of certain imaginary root spaces for rank 3 symmetric Kac-Moody algebras. We describe an explicit method for extracting combinatorics when the Dynkin diagram is bipartite (i.e. two of the nodes are not connected). As in rank two we believe these bounds are quite tight and we give computational evidence to this effect, although there is more error in rank 3 than in rank 2.