题目：Counting Representations and Kac Conjecture

报告人：Zongzhu Lin  教授

（Kansas State University）

报告时间：2013年7月9日（星期二）, 下午4:00-5:00

报告地点：管理科研楼1518室

摘要：
Given a finite quiver Q without loops, there is an associated Kac-Moody Lie algebra g. Kac proved that indecomposable finite dimensional representations of Q over a field have dimension vectors exactly being positive roots of g. Kac conjectured that the number of absolutely indecomposable representations with a fixed dimension factor /alpha over a finite field with q elements is a polynomial of q with non-negative integer coefficients and the constant term is exactly the dimension of the root space of g of the positive root /alpha. This conjecture has been proved recently by Hausel et al. In 1998, Jiuzhao Hua gave
an equivalent formulation to the conjecture in terms generating functions counting various representations. Hausel extends these generating functions and relates them to the Betti numbers of Nakajima quiver varieties. Hausel’s generalized Hua’s identity counts then number of rational points of various Nakajima quiver varieties over finite fields. Using Weil conjecture approach, these numbers are related to Betti numbers of the complex Nakajima quiver varieties. Nakajima’s theorem relates the top Betti numbers to the characters of the irreducible highest weight integrable representations of g. In this talk, I will outline the main ideas and possible equivariant variations. Hua’s generating functions also has a different extensions giving DT invariants.

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