Combinatorial realizations of crystals via torus actions on quiver varieties

Abstract

Let V(λ) be a highest-weight representation of a symmetric Kac-Moody algebra, and let B(λ) be its crystal. There is a geometric realization of B(λ) using Nakajima's quiver varieties. In many particular cases one can also realize B(λ) by elementary combinatorial methods. Here we study a general method of extracting combinatorial realizations from the geometric picture: we use Morse theory to index the irreducible components by connected components of the subvariety of fixed points for a certain torus action. We then discuss the case of ̂sln, where the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial realizations for each highest-weight crystal. In the case of B(Λ 0) we recover a family of realizations which was recently constructed by Fayers. This gives a more conceptual proof of Fayers' result as well as a generalization to higher level crystals. We also discuss a relationship with Nakajima's monomial crystal. © 2013 Springer Science+Business Media New York.