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Algorithmic aspects of semistability of quiver representations

Yuni Iwamasa, Taihei Oki, Tasuku Soma

TL;DR

...

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 $σ$-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 $σ$-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.

Algorithmic aspects of semistability of quiver representations

TL;DR

...

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 -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 -semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.
Paper Structure (40 sections, 36 theorems, 107 equations, 2 figures, 4 algorithms)

This paper contains 40 sections, 36 theorems, 107 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction and our contribution
  2. Semistability of quiver representations
  3. King's criterion
  4. Harder-Narasimhan filtration
  5. King's polyhedral cone, rank-one representations, and submodular flow
  6. Semistability of general quivers
  7. Remark.
  8. Our techniques
  9. $\sigma$-semistability.
  10. Maximizers in King's criterion.
  11. HN-filtration.
  12. Strongly polynomial-time algorithm for rank-one representations.
  13. Semistability of general quivers.
  14. Related work
  15. Organization of this paper
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $Q$ be an acyclic quiver, $V$ a representation of $Q$, and $\sigma$ a weight. There is a deterministic algorithm that decides the $\sigma$-semistability of $V$ in time polynomial in the size of $Q$, $\alpha(Q_0)$, bit complexity of $V$, and absolute values of the entries of $\sigma$.

Figures (2)

  • Figure 1: Generalized Kronecker quiver (left) and star quiver (right).
  • Figure 2: The left is an original quiver $Q$ and the right is the directed graph $D[V]$ constructed from $Q$ and a rank-one representation $V$ of $Q$. The red circles and blue squares in the right graph represent the vertices of $D[V]$ corresponding to the outgoing arcs (endowed with nonzero dual vector) and incoming arcs (endowed with nonzero vector) in $Q$, respectively.

Theorems & Definitions (59)

  • Theorem 1.1: informal version of \ref{['thm:ss']}
  • Example 1.2: nc-rank
  • Example 1.3: BL polytope
  • Theorem 1.4: informal version of \ref{['thm:King-maximizer']}
  • Theorem 1.5: informal version of \ref{['thm:HN']}
  • Theorem 1.6: informal version of \ref{['thm:submodular-flow', 'thm:rank-one:poly']}
  • Theorem 2.1: Franks2023
  • Theorem 2.2: Franks2018
  • Theorem 2.3: Theorem 1.13 in Burgisser2018a, specialized for operator scaling
  • Lemma 2.4
  • ...and 49 more