Algorithmic aspects of semistability of quiver representations

2024-07-09

arXiv.org

AI Summary

This paper addresses fundamental computational problems in quiver representation theory related to semistability: deciding semistability and σ-semistability, computing King-criterion maximizers, and constructing Harder–Narasimhan filtrations. Methodologically, it introduces submodular flow polyhedral theory into King’s stability framework—marking the first strong polynomial-time algorithm for σ-semistability testing of rank-one representations. It systematically characterizes the combinatorial structure of the King cone defined by King’s criterion and provides an explicit combinatorial encoding thereof in the rank-one case. Crucially, it uncovers intrinsic submodularity inherent in quiver representations, thereby establishing a computationally tractable foundation for geometric invariant theory. The proposed algorithms integrate linear programming, polyhedral combinatorics, and representation-theoretic techniques to uniformly resolve several long-standing computational challenges in quiver moduli theory.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Deciding semistability and σ-semistability of quiver representations
• Finding maximizers of King's criterion for quiver representations
• Computing Harder--Narasimhan filtration for quiver representations

Innovation

• Efficient algorithms for quiver semistability problems
• King cones encoded by submodular flow polytopes
• Submodularity in quiver representations