Duality between Final-Seed and Initial-Seed Mutations in Cluster Algebras

TL;DR

This paper explores the duality between final-seed and initial-seed mutations in cluster algebras, introducing $F$-matrices that exhibit self-duality similar to known matrix dualities, advancing understanding of mutation symmetries.

Contribution

It defines $F$-matrices for $F$-polynomials and demonstrates their self-duality, revealing a new symmetry in the mutation structure of cluster algebras.

Findings

$F$-matrices have self-duality.

Duality between mutations and initial-seed mutations established.

Enhanced understanding of mutation symmetries in cluster algebras.

Abstract

We study the duality between the mutations and the initial-seed mutations in cluster algebras, where the initial-seed mutations are the transformations of rational expressions of cluster variables in terms of the initial cluster under the change of the initial cluster. In particular, we define the maximal degree matrices of the $F$-polynomials called the $F$-matrices and show that the $F$-matrices have the self-duality which is analogous to the duality between the $C$- and $G$-matrices.