Structure of seeds in generalized cluster algebras

TL;DR

This paper extends the structure theory of seeds in generalized cluster algebras, providing formulas for cluster variables and coefficients in terms of c-vectors, g-vectors, and F-polynomials under certain conditions.

Contribution

It generalizes the seed structure theory from ordinary to generalized cluster algebras, including explicit formulas for key algebraic elements.

Findings

Formulas for cluster variables in terms of c-vectors, g-vectors, and F-polynomials

Extension of seed structure theory to generalized cluster algebras

Applicable under normalization and quasi-reciprocity conditions

Abstract

We study generalized cluster algebras introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasi-reciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster algebras by Fomin and Zelevinsky to generalized cluster algebras. As the main result, we obtain formulas expressing cluster variables and coefficients in terms of c-vectors, g-vectors, and F-polynomials.