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On the expansion formulas of cluster varieties from surfaces and their combinatorial properties

Vu Tung Lam Dinh, Ivan Chi-Ho Ip

Abstract

This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables arising from flips of a triangulation, corresponding to specific sequences of mutations. Our approach is grounded in a detailed combinatorial analysis over the standard $n$-triangulated $m$-gon (with explicit calculations for $n=1,2$). As a generalization, the non-simply-laced $G_2$ type is also considered. We prove the "well-triangulated" property for cluster mutations under flips, which provides a combinatorial framework for understanding the stability and transformation rules of these cluster algebra structures, and compute the monomial counts for the cluster expansion formula.

On the expansion formulas of cluster varieties from surfaces and their combinatorial properties

Abstract

This paper explores the cluster algebra structure of the moduli space of twisted -local systems on a surface. We derive general recurrence relations for cluster variables arising from flips of a triangulation, corresponding to specific sequences of mutations. Our approach is grounded in a detailed combinatorial analysis over the standard -triangulated -gon (with explicit calculations for ). As a generalization, the non-simply-laced type is also considered. We prove the "well-triangulated" property for cluster mutations under flips, which provides a combinatorial framework for understanding the stability and transformation rules of these cluster algebra structures, and compute the monomial counts for the cluster expansion formula.
Paper Structure (20 sections, 28 theorems, 161 equations, 31 figures)

This paper contains 20 sections, 28 theorems, 161 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cluster algebra
  4. Moduli spaces of $G$-local systems from surfaces
  5. Cluster realization of $\mathscr{A}_{G,\mathbb{S}}$ in type $A_n$
  6. Cluster expansion formulas for general flips
  7. Unpunctured surface case
  8. Combinatorial interpretation via stair paths
  9. Punctured surface case
  10. Solving recurrences
  11. Cluster expansion formulas of $n$-triangulated $m$-gon
  12. Notation and labeling conventions
  13. Well-triangulated Preservation and Number of Monomials Theorem
  14. General recursive formulas for particular values of $n$
  15. General number of terms property and examples
  16. ...and 5 more sections

Key Result

Theorem 1.1

For a triangulation $\mathcal{T}$ of a well-triangulated polygon $\mathcal{P}$ with the cluster realization of type $A_n$, after any sequence of flips, the resulting triangulation is also well-triangulated.

Figures (31)

  • Figure 1: The flip of a quadrilateral
  • Figure 2: Step-by-step construction of a flip in a $7$-triangulated quadrilateral
  • Figure 3: A $2$-triangulated triangle and the corresponding quiver
  • Figure 4: Original quadrilateral (left) and after flip (right)
  • Figure 5: Quivers corresponding to the quadrilaterals in Figure \ref{['fig:3']}
  • ...and 26 more figures

Theorems & Definitions (75)

  • Theorem 1.1: Well-triangulated preservation (Theorem \ref{['thm:well_triangulated_preservation']})
  • Theorem 1.2: Exponent Formula (Theorem \ref{['thm:exponents']})
  • Theorem 1.3: Number of monomials (Theorem \ref{['thm:monomial_count']})
  • Theorem 1.4: Recurrence Formulas for General Polygons (Theorems \ref{['thm:1tri']}, \ref{['thm:2tri']})
  • Corollary 1.5: Term Count Formulas (Corollaries \ref{['cor:diagonal-terms']}, \ref{['cor:inner-terms']})
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • ...and 65 more