A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers

Abstract

Let $M$ be a representation of an acyclic quiver $Q$ over an infinite field $k$. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of $M$. The algorithm is polynomial in the dimensions of $M$, the weights that induce the Harder–Narasimhan filtration of $M$, and the number of paths in $Q$. As a direct application, we also show that when $k$ is algebraically closed and when $M$ is unstable, the same algorithm produces Kempf’s maximally destabilizing one parameter subgroups for $M$.