Quantum F-polynomials in Classical Types

Abstract

In their "Cluster Algebras IV" paper, Fomin and Zelevinsky defined F-polynomials and g-vectors, and they showed that the cluster variables in any cluster algebra can be expressed in a formula involving the appropriate F-polynomial and g-vector. In "F-polynomials in Quantum Cluster Algebras," the predecessor to this paper, we defined and proved the existence of quantum F-polynomials, which are analogs of F-polynomials in quantum cluster algebras in the sense that cluster variables in any quantum cluster algebra can be expressed in a similar formula in terms of quantum F-polynomials and g-vectors. In this paper, we give formulas for both F-polynomials and quantum F-polynomials for cluster algebras of classical type when the initial exchange matrix is acyclic.