Harder-Narasimhan Filtration on Moment Map for Quiver Representations

TL;DR

This paper extends the understanding of the Harder-Narasimhan filtration in the context of quiver representations, linking geometric stratifications to algebraic invariants like Kac's polynomial.

Contribution

It generalizes existing methods to compute the dimensions of Harder-Narasimhan strata for quivers with many edges, revealing a connection to Kac's polynomial.

Findings

Computed dimensions of Harder-Narasimhan strata for complex quivers

Established a link between these dimensions and Kac's polynomial terms

Connected equivariant cohomology to the geometric stratification

Abstract

For a quiver without loops with many edges, we generalize the methods of Kac, Crawley-Boevey and Reineke and compute the dimension of Harder-Narasimhan strata of the zero set of the moment map. We notice a link between this dimension and the terms in the Kac's polynomial, which is given by the equivariant cohomology of this zero set.