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Quivers and moduli of their thin sincere representations in Macaulay2

Mary Barker, Patricio Gallardo

TL;DR

The paper introduces the ThinSincereQuivers Macaulay2 package to study acyclic quivers and their thin sincere representations through toric geometry. It develops the ToricQuiver datatype, enabling construction from incidence data or graphs, analysis of subquivers and stability, and computation of flow polytopes and the cone of weights, including wall-and-chamber decompositions. It connects flow polytopes to moduli spaces $\overline{M}(Q,\theta)$ via GIT, and provides tools to study reflexive polytopes, polytopal realizations, and toric compactifications of moduli spaces such as $M_{0,n}$. The work offers concrete algorithms and examples that bridge quiver representations, polyhedral geometry, and toric moduli, with practical implications for mirror symmetry and toric geometry.

Abstract

We introduce the Macaulay2 package ThinSincereQuivers for studying acyclic quivers, the moduli of their thin-sincere representations, and the reflexive flow polytopes associated to them. We provide some background on the topic and illustrate how the package recovers examples from the literature.

Quivers and moduli of their thin sincere representations in Macaulay2

TL;DR

The paper introduces the ThinSincereQuivers Macaulay2 package to study acyclic quivers and their thin sincere representations through toric geometry. It develops the ToricQuiver datatype, enabling construction from incidence data or graphs, analysis of subquivers and stability, and computation of flow polytopes and the cone of weights, including wall-and-chamber decompositions. It connects flow polytopes to moduli spaces via GIT, and provides tools to study reflexive polytopes, polytopal realizations, and toric compactifications of moduli spaces such as . The work offers concrete algorithms and examples that bridge quiver representations, polyhedral geometry, and toric moduli, with practical implications for mirror symmetry and toric geometry.

Abstract

We introduce the Macaulay2 package ThinSincereQuivers for studying acyclic quivers, the moduli of their thin-sincere representations, and the reflexive flow polytopes associated to them. We provide some background on the topic and illustrate how the package recovers examples from the literature.

Paper Structure

This paper contains 10 sections, 2 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Description
  4. Constructing quivers
  5. Subquivers and their stability
  6. Regular flows and cone of weights
  7. Walls and chambers on the space of weights
  8. Applications
  9. To polytopes
  10. Applications to moduli spaces

Key Result

Lemma 3.1

hille2003quivers Each wall $W$ in $C(Q)$ is contained in a hyperplane of the form where $Q= Q_0^+ \sqcup Q_0^-$ and the full subquivers $Q^+$ and $Q^-$ with vertices $Q_0^+$ and $Q_0^-$ are connected.

Figures (3)

  • Figure 1: Examples of acyclic quivers
  • Figure 2: Subquivers of the bipartite quiver as defined in Example \ref{['ex:defQuiver']}. They are labeled by the subsets $I= \{0,1,2,4 \}$ and $\{0,1,4,5 \}$
  • Figure 3: Cone of weights $C(Q)$ (right), polytope $\Delta(\theta)$ associated to the weight $\theta = (-2,1,-1,2)$ (center) and $\theta$-stable trees parametrized by the vertices of the polytope $\Delta(\theta)$ (left). Here, the quiver $Q$ is constructed from the complete graph with four vertices.

Theorems & Definitions (3)

  • Example 2.1
  • Lemma 3.1
  • Theorem 4.1