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

Abstract

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.

Stratifications Associated to Reductive Group Actions on Affine Spaces

“…However we provide a concise proof of this fact for completeness of the paper. Whilst this paper was being completed, we note that Zamora has also given a proof that the Kempf filtration (this is a natural filtration associated to an adapted 1-PS) is equal to the Harder-Narasimhan filtration for quiver representations [18].…”

A polynomial time algorithm for Harder-Narasimhan filtrations of representations of acyclic quivers

“…The relation between (2) and ( 3) is already unraveled by the paper [Zam14]. The main result is that for an unstable representation, its Harder-Narasimhan filtration coincides with its Kempf filtration.…”

“…First, computing Kempf's one parameter subgroups in general is very hard. If a version of Theorem B exists over the field of complex numbers, then the result of [Zam14] ensures that Kempf's one parameter subgroups can be computed efficiently for representations of bipartite quivers. What is more, [Hos14] also shows that the stratification of the space of representations of a quiver of a fixed dimension by Harder-Narasimhan types coincides with the stratification by Kempf's one parameter subgroups.…”

“…The discussion with him and questions raised by him offered many insights that fostered results in the paper. The author also gives thanks to Alfonso Zamora to kindly answered the author's questions on certain results in the paper [Zam14].…”

Harder–Narasimhan filtration for rank 2 tensors and stable coverings

“…For example, in [14], a similar correspondence is proven for representations of a finite quiver in the category of finite dimensional vector spaces over an algebraically closed field of arbitrary characteristic. For tensors in general, a notion of Harder-Narasimhan filtration is unknown, and rank 2 tensors is a particular example of tensors, for which this method can be implemented.…”