Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some coefficients of rank 2 cluster scattering diagrams

Ryota Akagi

TL;DR

The paper addresses translating rank-2 cluster scattering diagrams from a dilogarithm-identity framework into formal power-series language to access explicit wall-coefficient data. It derives an explicit wall-function formula $f_{\bf n_0}$ in terms of $U_{k{\bf n_0}}(c,b)$ and a gcd-based factor, and provides a translation from the dilogarithm exponents $u_{\bf n}(c,b)$ to scattering coefficients $\tau^{(b,c)}(k{\bf n_0})$. Using this translation, it proves key conjectures from ERS24 about polynomiality, degree bounds, and positivity of the wall-coefficient polynomials, including a complete proof of Conjecture 11 with a positive binomial expansion. The results illuminate the combinatorial structure of rank-2 walls, connect dilogarithm identities to formal series, and complement subsequent approaches via quiver moduli and tight gradings. Overall, the work provides explicit, testable expressions for wall data in rank-2 cluster scattering diagrams and validates several conjectures on their arithmetic structure.

Abstract

The purpose of this paper is to translate the expression of rank 2 cluster scattering diagrams via dilogarithm elements into via formal power series. As a corollary, we prove some conjectures introduced by Thomas Elgin, Nathan Reading, and Salvatore Stella.

Some coefficients of rank 2 cluster scattering diagrams

TL;DR

The paper addresses translating rank-2 cluster scattering diagrams from a dilogarithm-identity framework into formal power-series language to access explicit wall-coefficient data. It derives an explicit wall-function formula in terms of and a gcd-based factor, and provides a translation from the dilogarithm exponents to scattering coefficients . Using this translation, it proves key conjectures from ERS24 about polynomiality, degree bounds, and positivity of the wall-coefficient polynomials, including a complete proof of Conjecture 11 with a positive binomial expansion. The results illuminate the combinatorial structure of rank-2 walls, connect dilogarithm identities to formal series, and complement subsequent approaches via quiver moduli and tight gradings. Overall, the work provides explicit, testable expressions for wall data in rank-2 cluster scattering diagrams and validates several conjectures on their arithmetic structure.

Abstract

The purpose of this paper is to translate the expression of rank 2 cluster scattering diagrams via dilogarithm elements into via formal power series. As a corollary, we prove some conjectures introduced by Thomas Elgin, Nathan Reading, and Salvatore Stella.

Paper Structure

This paper contains 12 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Structure of this paper
  5. Definition
  6. From dilogarithm identities
  7. Formal power series expression
  8. Translation into the scattering coefficients
  9. Proof of some conjectures in ERS24
  10. Conjecture 1, 2, 3, 5, 6
  11. Conjecture 11
  12. Conjecture 15 and related results of Conjecture 14, 16,17,18

Key Result

Theorem 1.2

Consider the cluster scattering diagram $\mathfrak{D}_{b,c}$ with the initial exchange matrix $B=\left(\right)$($b,c \in \mathbb{Z}_{\geq 1}$). Then, for each ${\bf n}_0=(n_1,n_2) \in \mathbb{Z}^{2}_{\geq 0}\setminus\{(0,0)\}$ with $\gcd(n_1,n_2)=1$, the wall function $f_{{\bf n}_0}$ corresponding t where $g({\bf n}_0;b,c)=\frac{\gcd(n_1b,n_2,c)}{\gcd(n_1,n_2)}$ and $U_{k{\bf n}_0}(c,b) \in \mathb

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 1.2: Theorem \ref{['thm: expression of wall function']}
  • Proposition 1.3: Proposition \ref{['prop: 1,2,3']}, \ref{['prop: 5,6']}, \ref{['prop: 14,15,16,17,18']}
  • Remark 1.4
  • Definition 2.1
  • Proposition 2.2: Nak23
  • Proposition 2.3: Nak23
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: Nak23 Ordering lemma
  • ...and 27 more