University of Toronto's Symplectic Geometry Seminar

Abstract: In this talk I will discuss a recent work (arXiv:math.RT/0310314) on the relation between two realizations of crystal graphs. Crystal graphs, which can be viewed as the q=0 limit of quantum groups, reduce many questions in representation theory (such as computation of characters and decomposition of tensor products of representations into sums of indecomposable ones) to combinatorics. Crystal graphs of representations of Kac-Moody algebras can be realized geometrically on the set of irreducible components of certain varieties attached to quivers as well as on combinatorial objects such as Young tableaux and Young walls. I will discuss an explicit isomorphism between these two constuctions. Some benefits of this relationship include obtaining an explicit enumeration of irreducible components of quiver varieties by classical combinatorial objects as well as giving a geometric interpretation of the combinatorial constructions (which allows us, in some cases, to extend the combinatorial constructions to more general cases). I will review the necessary material on crystal graphs and quiver varieties.