Algorithmic aspects of semistability of quiver representations

TL;DR

This paper develops efficient algorithms for determining semistability of quiver representations, including rank-one cases, using submodularity and polyhedral cone analysis, advancing computational methods in representation theory.

Contribution

It introduces new algorithms for semistability problems in quiver representations, especially leveraging submodular flow polytopes for rank-one cases, and explores the structure of King cones.

Findings

Deciding semistability and $\sigma$-semistability can be done efficiently.

Rank-one representations' King cones are encoded by submodular flow polytopes.

Strongly polynomial time algorithms are achieved for certain cases.

Abstract

We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and $\sigma$-semistability, finding the maximizers of King's criterion, and computing the Harder--Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King's criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the $\sigma$-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.