Duality between Final-Seed and Initial-Seed Mutations 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.

References in corpus (2)

• Initial-seed recursions and dualities for d-vectors
• Uniformly column sign-coherence and the existence of maximal green sequences

Cited by in corpus (6)

• Combinatorial cluster expansion formulas from triangulated surfaces
• Dual canonical bases and quantum cluster algebras
• The valuation pairing on an upper cluster algebra
• Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2
• Earthquake theorem for cluster algebras of finite type
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams