  We define a generalized version of the frieze variety introduced by Lee, Li,
Mills, Seceleanu and the second author. The generalized frieze variety is an
algebraic variety determined by an acyclic quiver and a generic specialization
of cluster variables in the cluster algebra for this quiver. The original
frieze variety is obtained when this specialization is (1, . . . , 1). The main
result is that a generalized frieze variety is determined by any generic
element of any component of that variety. We also show that the "Coxeter
mutation" cyclically permutes these components. In particular, this shows that
the frieze variety is invariant under the Coxeter mutation at a generic point.
The paper contains many examples which are generated using a new technique
which we call an invariant Laurent polynomial. We show that a symmetry of a
mutation of a quiver gives such an invariant rational function.
