Fractal phenomenon in $c$- and $g$-vectors of the Markov quiver

TL;DR

This paper explores the fractal structure of $c$- and $g$-vectors in rank 3 skew-symmetrizable matrices, revealing recursive formulas and a parameterization by coprime integers, with applications to the $G$-fan.

Contribution

It introduces explicit recursive formulas for modified $c$- and $g$-vectors and uncovers their fractal and recursive structure in the context of Markov quivers.

Findings

Identifies fractal patterns in $G$-fans of Markov-type cluster algebras.

Provides recursive formulas for $c$- and $g$-vectors based on coprime integer pairs.

Describes the recursive generation of connected components outside the $G$-fan support.

Abstract

We study the $C$- and $G$-patterns associated with rank $3$ skew-symmetrizable matrices of $B$-invariant type, including the Markov quiver. Motivated by the self-contained simple mutations in Markov-type cluster algebras, we prove that large classes of subpatterns of modified $c$- and $g$-vectors are linearly isomorphic, yielding a fractal structure of the corresponding $G$-fan. We further derive explicit recursive formulas for all modified $c$- and $g$-vectors in terms of integer pairs satisfying a recursion analogous to the Calkin-Wilf tree, which leads to a parameterization by coprime integers. As an application, we describe all connected components of the complement of the support of the $G$-fan, and show that they are generated recursively by three kinds of linear maps.