### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 040, 24 pages      arXiv:1808.02156      https://doi.org/10.3842/SIGMA.2019.040

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

Shogo Fujiwara and Yasuaki Gyoda
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan

Received October 16, 2018, in final form May 10, 2019; Published online May 15, 2019

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.

Key words: cluster algebra; mutation; duality.

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