On the Harder-Narasimhan filtration for finite dimensional representations of quivers

TL;DR

This paper establishes that the Harder-Narasimhan filtration for unstable finite-dimensional quiver representations aligns with the maximal destabilizing 1-parameter subgroup from Geometric Invariant Theory, linking stability notions to GIT.

Contribution

It proves the equivalence between the Harder-Narasimhan filtration and Kempf's maximal destabilizing subgroup for quiver representations.

Findings

Harder-Narasimhan filtration matches Kempf's 1-parameter subgroup.

Provides a GIT interpretation of the Harder-Narasimhan filtration.

Links stability concepts with geometric invariant theory in quiver moduli.

Abstract

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.