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

Cheng, Chi-Yu

doi:10.2140/ant.2024.18.319

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

Chi-Yu Cheng

Vol. 18 (2024), No. 2, 319–347

DOI: 10.2140/ant.2024.18.319

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 $M$ .

Keywords

geometric invariant theory, weight stability, slope stability, representation, quiver, Harder–Narasimhan filtration, discrepancy

Mathematical Subject Classification

Primary: 14L24, 14Q20, 16G20

Milestones

Received: 24 June 2022

Revised: 16 January 2023

Accepted: 20 March 2023

Published: 6 February 2024

Authors

Chi-Yu Cheng
University of Missouri Columbia, MO United States

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Vol. 18, No. 2, 2024