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Cluster scattering diagrams via quiver moduli and tight gradings

Amanda Burcroff, Kyungyong Lee, Lang Mou, Gregg Musiker, Markus Reineke

TL;DR

The paper advances rank-2 cluster scattering diagrams by marrying quiver-moduli techniques with the recently developed tight-gradings framework. It proves a substantial set of Elgin–Reading–Stella conjectures about wall-function coefficients and provides a novel, geometry-driven proof of Weyl group symmetry via mutation and retraction on tight gradings. Central to the approach is the change-of-lattice mechanism, which links $ rak{D}_{(b,c)}$ and its swapped form, while wall-functions are computed through Euler characteristics of quiver moduli spaces, yielding explicit polynomial and positive-formulas. The work also introduces a vivid footprint-tiling picture for tight gradings, enabling combinatorial counting and offering pathways to deeper connections with stable trees and asymptotic growth of wall-functions.

Abstract

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a collection of conjectures posed by Elgin--Reading--Stella concerning the structural and enumerative properties of the wall-function coefficients. The tight grading perspective also provides a new proof of the Weyl group symmetry of the scattering diagram.

Cluster scattering diagrams via quiver moduli and tight gradings

TL;DR

The paper advances rank-2 cluster scattering diagrams by marrying quiver-moduli techniques with the recently developed tight-gradings framework. It proves a substantial set of Elgin–Reading–Stella conjectures about wall-function coefficients and provides a novel, geometry-driven proof of Weyl group symmetry via mutation and retraction on tight gradings. Central to the approach is the change-of-lattice mechanism, which links and its swapped form, while wall-functions are computed through Euler characteristics of quiver moduli spaces, yielding explicit polynomial and positive-formulas. The work also introduces a vivid footprint-tiling picture for tight gradings, enabling combinatorial counting and offering pathways to deeper connections with stable trees and asymptotic growth of wall-functions.

Abstract

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a collection of conjectures posed by Elgin--Reading--Stella concerning the structural and enumerative properties of the wall-function coefficients. The tight grading perspective also provides a new proof of the Weyl group symmetry of the scattering diagram.

Paper Structure

This paper contains 25 sections, 22 theorems, 103 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Elgin--Reading--Stella's conjectures
  4. Main results
  5. Rank 2 scattering diagrams and the change-of-lattice
  6. Wall-functions via quiver moduli
  7. Tight gradings
  8. Maximal Dyck paths
  9. Definition of tight gradings
  10. Structure of Tight Gradings
  11. A pictorial description of tight gradings via tiling
  12. Frames
  13. Construction of footprints associated to $u_r^{(s)}$
  14. Construction of footprints associated to $v_{r}^{(s)}$
  15. Reinterpretation of tight gradings
  16. ...and 10 more sections

Key Result

Theorem 1.2

Figures (7)

  • Figure 1: $\mathcal{P}(2d_1, 2d_2)$
  • Figure 2: A frame $L_\mathfrak{p}$.
  • Figure 3: An illustration of the proof of \ref{['lem: b c degree tight gradings']}. Following the construction, the remaining red and dark red tiles are contained in the red shaded region, while the remaining cyan and dark blue tiles are contained in the cyan shaded region.
  • Figure 4: Tight gradings in the case $j=5$ with $\mathbf{P}_1 = (1,1)$.
  • Figure 5: Tight gradings in the case $j=5$ with $\mathbf{P}_1 = (2)$.
  • ...and 2 more figures

Theorems & Definitions (75)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 2.1
  • Proposition 2.2: Change-of-lattice
  • Corollary 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.3
  • ...and 65 more