Program
Kac's conjecture and quiver varieties
Student No.:50
Time:Wed 14:30-16:30
Instructor:Fan Xu,Fan Qin  
Place:Seminar Room, Floor 3, Jin Chun Yuan West Building
Starting Date:2013-3-6
Ending Date:2013-6-5

 

Description:

 

 

Kac proved that the number of absolutely indecomposable quiver representations over a finite field is given by the evaluation of an integer polynomial at the field's cardinality. He conjectured that this polynomial has non-negative coefficients.

 

Recently, Letellier, Hausel, and Rodriguez-Villegas proved this conjecture for general cases. Their method consists of realizing this polynomial via certain quiver varieties. As a byproduct, they also obtained a new proof of the positivity of certain refined Donaldson-Thomas invariants.

 

In this seminar, we shall focus on the part of Kac's conjecture.

 

Plan of content:

 (1) Kac's conjecture and its proof: an introduction

 (2) Symmetric functions and Hua's formula ([Hua])

 (3) The proof of Kac's theorem ([A. Hubery])

 (4) Introduction to quiver varieties (section 2.1)

 (5) Weyl group action on the cohomology groups of quiver varieties (section 2.2.1)

 (6) If possible: schemes, sheaf functors, and etale-chomology

 (7) Point counting via Grothendieck's trace formula (section 2.2.2)

 (8) Point counting via the Fourier transform (section 2.3)

 (9) Main theorem: cohomological interpretation of Kac's polynomials (section 2.4, 2.5)

 (10) Schiffmann's approach via Lusztig nilpotent varieties ([Schiffmann])

 (11) If possible: relation to DT-invariants (section 3)

 

Reference:

 

Tamas Hausel, Emmanuel Letellier, Fernando Rodriguez-Villegas, Positivity of Kac polynomials and DT-invariants for quivers,

arXiv:1204.2375