Let \(Q\) be an acyclic quiver and \(X_?\colon\mathcal C_Q\to\mathbb Z[\mathbf u^{\pm 1}]\) be the associated Caldero-Chapoton map. The main results of the paper are formulas for \(X_M\) and \(X_MX_N\) when both \(M,N\) are indecomposable regular modules.
A family of polynomials \(\{P_n\}_{n\geq 0}\) are introduced by a recursion relation; these generalize the classical Chebyshev polynomials of the second kind. Numerous examples are given for the reader’s convenience. The first main result (Cor. 4.2) says that a cluster algebra \(\mathcal A\) of Dynkin type \(\mathbb A_r\) can be presented as \(\mathcal A\simeq\mathbb Z[t_0,\dots,t_r]/(P_{r+1}(t_0,\dots,t_r)-1)\).
The next main result (Theorem 5.1) gives, when \(M\) is an indecomposable regular representation of any acyclic quiver, a formula for \(X_M\) in terms of these generalized Chebyshev polynomials evaluated at the quasi-composition factors of \(M\). In particular, this implies that \(X_M\) belongs to the associated cluster algebra when the quasi-composition factors are rigid.
Finally, Theorems 7.2 and 8.3 give multiplication formulas for cluster characters associated to indecomposable regular modules in affine and wild type, respectively. The formulas take the form \(X_MX_N=X_AX_B+X_CX_D\) where \(M,N\) are indecomposable modules in the same regular component of the AR quiver of \(Q\), and \(A,B,C,D\) are indecomposable modules in the same regular component and everything is explicitly given in terms of quasi-socles and quasi-length. In the affine case, the product admits an interpretation as a Hall product in the cluster category \(\mathcal C_Q\) (Corollary 7.1).