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Positivity of generalized cluster scattering diagrams

Amanda Burcroff, Kyungyong Lee, Lang Mou

TL;DR

This work develops a comprehensive positivity theory for generalized cluster scattering diagrams by introducing tight gradings and shadowed gradings, which yield manifestly positive wall-functions in rank 2 and extend to higher ranks. It connects wall-function positivity to the Laurent phenomenon and strong positivity for Chekhov–Shapiro generalized cluster algebras, and links these combinatorial structures to geometric invariants, including Euler characteristics of quiver moduli spaces and relative Gromov–Witten invariants. The authors prove a tight-grading formula, show the generalized greedy basis equals the generalized theta basis, and establish a robust cluster-complex structure that underpins universality and atomicity of theta functions. Collectively, these results provide a positive, combinatorially transparent framework for generalized cluster algebras with polynomial exchange relations and their geometric and representation-theoretic applications.

Abstract

We introduce a new class of combinatorial objects, named tight gradings, which are certain nonnegative integer-valued functions on maximal Dyck paths. Using tight gradings, we derive a manifestly positive formula for any wall-function in a rank-2 generalized cluster scattering diagram. We further prove that any consistent rank-2 scattering diagram is positive with respect to the coefficients of initial wall-functions. Moreover, our formula yields explicit expressions for relative Gromov-Witten invariants on weighted projective planes and the Euler characteristics of moduli spaces of framed stable representations on complete bipartite quivers. Finally, by leveraging the rank-2 positivity, we show that any higher-rank generalized cluster scattering diagram has positive wall-functions, which leads to a proof of the positivity of the Laurent phenomenon and the strong positivity of Chekhov-Shapiro's generalized cluster algebras.

Positivity of generalized cluster scattering diagrams

TL;DR

This work develops a comprehensive positivity theory for generalized cluster scattering diagrams by introducing tight gradings and shadowed gradings, which yield manifestly positive wall-functions in rank 2 and extend to higher ranks. It connects wall-function positivity to the Laurent phenomenon and strong positivity for Chekhov–Shapiro generalized cluster algebras, and links these combinatorial structures to geometric invariants, including Euler characteristics of quiver moduli spaces and relative Gromov–Witten invariants. The authors prove a tight-grading formula, show the generalized greedy basis equals the generalized theta basis, and establish a robust cluster-complex structure that underpins universality and atomicity of theta functions. Collectively, these results provide a positive, combinatorially transparent framework for generalized cluster algebras with polynomial exchange relations and their geometric and representation-theoretic applications.

Abstract

We introduce a new class of combinatorial objects, named tight gradings, which are certain nonnegative integer-valued functions on maximal Dyck paths. Using tight gradings, we derive a manifestly positive formula for any wall-function in a rank-2 generalized cluster scattering diagram. We further prove that any consistent rank-2 scattering diagram is positive with respect to the coefficients of initial wall-functions. Moreover, our formula yields explicit expressions for relative Gromov-Witten invariants on weighted projective planes and the Euler characteristics of moduli spaces of framed stable representations on complete bipartite quivers. Finally, by leveraging the rank-2 positivity, we show that any higher-rank generalized cluster scattering diagram has positive wall-functions, which leads to a proof of the positivity of the Laurent phenomenon and the strong positivity of Chekhov-Shapiro's generalized cluster algebras.

Paper Structure

This paper contains 43 sections, 67 theorems, 324 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formula for wall-functions
  3. Positivity of scattering diagrams
  4. Positivity of generalized cluster algebras
  5. Structure of the paper
  6. Rank 2 scattering diagrams and theta functions
  7. Scattering diagrams in rank 2
  8. Generalized cluster scattering diagrams
  9. Broken lines and theta functions
  10. Theta functions are universal Laurent
  11. Greedy bases for rank-2 generalized cluster algebras
  12. Generalized greedy elements
  13. Compatible gradings
  14. Generalized greedy element formula
  15. The generalized greedy basis equals the generalized theta basis
  16. ...and 28 more sections

Key Result

Theorem 1.1

Fix coprime positive integers $(a,b)$. For each $k \geq 1$, choose integers $(d_{1,k},d_{2,k})$ such that $ad_{2,k} - bd_{1,k} = \pm 1$, $ka \leq d_{1,k}$, and $kb \leq d_{2,k}$. Then where the second sum is over all tight gradings $\omega$ on the Dyck path $\mathcal{P}(d_{1,k}, d_{2,k})$ of total horizontal weight $kb$ and total vertical weight $ka$, and $\mathrm{wt}(\omega)\in \Bbbk\llbracket t

Figures (8)

  • Figure 1: $\mathop{\mathrm{Scat}}\nolimits(P_1, P_2)$, where $P_1 = 1 + p_{1,1}x + p_{1,2}x^2 + p_{1,3}x^3$ and $P_2 = 1 + p_{2,1}y$.
  • Figure 2: In the top images, we depict gradings $\omega_1$ and $\omega_2$ on the Dyck paths $\mathcal{P}(6,4)$ and $\mathcal{P}(7,4)$ from \ref{['main_exmp']}, where edges with no weight shown are assigned weight $0$. Below, we depict the corresponding rectangular tilings. The grading $\omega_1$ has overlapping rectangles and hence is not compatible, while the grading $\omega_2$ is.
  • Figure 3: The three tight gradings in \ref{['tight_exmp']}(2)(3)(4).
  • Figure 4: The four tight gradings in the setting of \ref{['exmp: affine 2 2 gradings']} when $k = 3$.
  • Figure 5: The five tight gradings in the setting of \ref{['exmp: affine 4 1 gradings']} when $k = 2$.
  • ...and 3 more figures

Theorems & Definitions (193)

  • Theorem 1.1: Tight grading formula
  • Theorem 1.2: \ref{['thm: full positivity rk 2']}
  • Theorem 1.3: \ref{['thm: positive wall-function']}
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 183 more