Companion cluster algebras to a generalized cluster algebra

Abstract

We study the $c$-vectors, $g$-vectors, and $F$-polynomials for generalized cluster algebras satisfying a normalization condition and a power condition recovering classical recursions and separation of additions formulas. We establish a relationship between the $c$-vectors, $g$-vectors, and $F$-polynomials of such a generalized cluster algebra and its (left- and right-) companion cluster algebras. Our main result states that the cluster variables and coefficients of the (left- and right-) companion cluster algebras can be recovered via a specialization of the $F$-polynomials.

14 pages, 4 tables

References in corpus (3)

• Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras
• Structure of seeds in generalized cluster algebras
• Quantum affine algebras at roots of unity and generalised cluster algebras

Cited by in corpus (4)

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• Snake Graphs from Triangulated Orbifolds
• On some combinatorial properties of generalized cluster algebras
• Symplectic groupoids for cluster manifolds