Abstract
We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type \( \mathbb{A} \) and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type \( \mathbb{A} \). We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.
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Dupont, G. Cluster Multiplication in Regular Components via Generalized Chebyshev Polynomials. Algebr Represent Theor 15, 527–549 (2012). https://doi.org/10.1007/s10468-010-9248-0
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DOI: https://doi.org/10.1007/s10468-010-9248-0
Keywords
- Cluster algebras
- Affine quivers
- Wild quivers
- Regular components
- Cluster multiplication
- Orthogonal polynomials