A quiver variety approach to root multiplicities

TL;DR

This paper introduces combinatorial upper bounds for the dimensions of imaginary root spaces in symmetric Kac-Moody algebras using quiver varieties, providing explicit and accurate bounds in rank two cases.

Contribution

It develops a new approach linking quiver varieties to root multiplicities, with explicit bounds in rank two that are often exact.

Findings

Bounds are accurate and often exact for large roots in rank two.

The framework generalizes to symmetric Kac-Moody algebras.

Explicit bounds are provided for specific cases.

Abstract

We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. The framework is general, but we only work out specifics in rank two. In that case we give explicit bounds. These turn out to be quite accurate, and in many cases exact, even for some fairly large roots.