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.
Tingley, Peter. A Quiver Variety Approach to Root Multiplicities. Algebraic Combinatorics, 4, 1: 163-174, 2021. Retrieved from Loyola eCommons, Mathematics and Statistics: Faculty Publications and Other Works, http://dx.doi.org/10.5802/alco.158
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