Math 7310: Topics in Algebra -- Quiver varieties
The course description:
Prof. Allen Knutson
MWF 12:20 in Malott 205
"Geometric representation theory" is misnamed; it is a collection of examples of groups acting on the homology of various varieties. One of the two best examples of such actions is Nakajima's 1994 action of a Lie group on the top homology groups of "quiver varieties", where the Lie group and the varieties are both defined from a (rather generalized) Dynkin diagram. This prompts the question of what should act on the total homology or K-theory of quiver varieties, and it turns out to be the corresponding "quantized loop algebra".
The principal topics in this course, in order, will be
I am writing a book on the subject, the first 42 pp here (updated 10/29):
Such representations had already appeared in statistical mechanics, where they provide tools for "completely integrating" some interesting stat mech problems. The connection of these "R-matrices" to the quiver variety geometry was spelled out in 2012 by Maulik and Okounkov. In the last couple of years Zinn-Justin and I have been using these tools to solve old problems in the cohomology rings of Grassmannians and other flag manifolds, a circle of problems called "Schubert calculus".
All of this will be built from the ground up. I will assume basic knowledge of homology and cohomology (though equivariant cohomology and K-theory would be nice too), and some exposure to Lie groups and their representations. Some results in algebraic geometry will be taken on faith.