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

TL;DR

This paper presents a polynomial-time deterministic algorithm for computing the Harder-Narasimhan filtration of representations of acyclic quivers, with applications to identifying destabilizing subgroups in algebraically closed fields.

Contribution

It introduces a new efficient algorithm for Harder-Narasimhan filtrations and applies it to find Kempf's destabilizing subgroups in algebraically closed fields.

Findings

Algorithm runs in polynomial time relative to input size.

Successfully computes Harder-Narasimhan filtrations for quiver representations.

Identifies Kempf's maximally destabilizing subgroups using the same algorithm.

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 $M$.