• quiver representations quiver varieties

Given a finite quiver Q and a dimension vector alpha for it, Kac has shown in 1980 that there exists a polynomial a_{Q,alpha}(t) with integer coefficients such that it has the following counting property:
a_{Q,alpha}(p^r)={number of isomorphism classes of absolutely indecomposable representations of Q of dimension vector alpha over the finite field of cardinality p^r}. A representation V of Q over a field k is said to be absolutely indecomposable if its base change over the algebraic closure of k is indecomposable.

Kac conjectured two properties of these objects: the conjecture studied in the thesis stated that these polynomials should have had nonnegative coefficients. In this thesis it is worked out a simpler case of the proof of this statement, regarding just indivisible dimension vectors, i.e when the greatest common divisor of the components of alpha is equal to 1. This case was worked out by Crawley-Boevey and Van der Bergh in 2004.

The proof of the conjecure is obtained by interpretating the coefficients of a_{Q,alpha} as the dimension of the cohomology groups of certain algebraic varieties associated to Q. These are called quiver varieties and are roughly obtained by considering moduli space of quiver representations. The third chapter of the thesis is actually dedicated to building up such geometrical objects.