# University of Toronto's Symplectic Geometry Seminar

Jan. 26, 2004, 2:10 - 3

SS5017A

## Alistair Savage

###
University of Toronto

##
"Quiver Varieties, Young Tableaux and Crystal Bases"

** 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.