Generalized Chebyshev Polynomials and Regular Components of Auslander-Reiten Quivers

Greoire Dupont

Caldero and Zelevinsky introduced normalized Chebyshev polynomials of the second kind in order to study the semi-canonical basis of a cluster algebra associated to the Kronecker quiver. We introduce a generalization of these polynomials arising naturally in the context of acyclic cluster algebras of infinite types and cluster algebras associated to equioriented quivers of Dynkin type A. These polynomials are used in order to describe values of the Caldero-Chapoton map associated to regular modules over the path algebra of an acyclic quiver. In particular, they allow to prove cluster multiplication formulas for regular modules over the path algebra of an affine quiver.