Combinatorial realizations of crystals via torus actions on quiver varieties

Sam, Steven V. ; Tingley, Peter

Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 271-300.

Résumé

Let $V(\lambda)$ be a highest-weight representation of a symmetric Kac-Moody algebra, and let $B(\lambda)$ be its crystal. There is a geometric realization of $B(\lambda)$ using Nakajima's quiver varieties. In many particular cases one can also realize $B(\lambda)$ 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 $\widehat{\mathfrak{sl}}_{n}$, 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 $B(\varLambda_{0})$ 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.

Détail

Classification : 05E10, 05C90
Keywords: crystals, partitions, quiver varieties, torus actions