Non-commutative Rank and Semi-stability of Quiver Representations

2021/10/29 by Alana Huszar, Huszar, Alana
Computer Science · Mathematics · #16G20 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Complexity and Algorithms in Graphs #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2111.00039

openalex publication_date 2021/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fortin and Reutenauer defined the non-commutative rank for a matrix with entries that are linear functions. The non-commutative rank is related to stability in invariant theory, non-commutative arithmetic circuits, and Edmonds' problem. We will generalize the non-commutative rank to the representation theory of quivers and define non-commutative Hom and Ext spaces. We will relate these new notions to King's criterion for σ-stability of quiver representations, and the general Hom and Ext spaces studied by Schofield. We discuss polynomial time algorithms that compute the non-commutative Homs and Exts and find an optimal witness for the σ-semi-stability of a quiver representation.

Related

• Algorithmic aspects of semistability of quiver representations
• Edmonds' problem and the membership problem for orbit semigroups of quiver representations
• Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry
• Non-commutative Edmonds' problem and matrix semi-invariants
• Representations of marked quivers