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Finite type and completeness of $g$-fans

Toshiya Yurikusa

TL;DR

The paper establishes that a skew-symmetrizable matrix $B$ is of finite type if and only if its $g$-fan $\\mathcal{F}(B)$ is complete and equivalently its support contains all lattice points in $\\mathbb{Z}^n$. It uses scattering diagrams and a pull-back construction to extend known results from the skew-symmetric case to the general skew-symmetrizable setting, proving that the absence of lattice points outside $|\\mathcal{F}(B)|$ forces finite type. The argument reduces to rank-2 submatrices and leverages mutations of $g$-vectors to connect geometric chambers with algebraic mutations. This work links cluster-algebra combinatorics, the geometry of fans, and representation-theoretic methods, providing a complete geometric criterion for finite type.

Abstract

We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.

Finite type and completeness of $g$-fans

TL;DR

The paper establishes that a skew-symmetrizable matrix is of finite type if and only if its -fan is complete and equivalently its support contains all lattice points in . It uses scattering diagrams and a pull-back construction to extend known results from the skew-symmetric case to the general skew-symmetrizable setting, proving that the absence of lattice points outside forces finite type. The argument reduces to rank-2 submatrices and leverages mutations of -vectors to connect geometric chambers with algebraic mutations. This work links cluster-algebra combinatorics, the geometry of fans, and representation-theoretic methods, providing a complete geometric criterion for finite type.

Abstract

We study the -fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its -fan is complete; equivalently (as we show), its support contains all lattice points.
Paper Structure (3 sections, 8 theorems, 8 equations, 1 figure)

This paper contains 3 sections, 8 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mutations of matrices and $g$-vector tuples
  3. Pull back of scattering diagrams

Key Result

Theorem 1.1

For a skew-symmetrizable matrix $B$, the following are equivalent:

Figures (1)

  • Figure 1: Examples of the $g$-vector fans $\mathcal{F}(B_{b,c})$

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: FZ03
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8: Re14
  • Theorem 2.9: GHKK18NZ12
  • ...and 7 more