  Let V be a unitary space. Suppose G is a subgroup of the full symmetric group
S_m and X is an irreducible unitary representation of G. In this paper, we
introduce the generalized Cartesian symmetry class over V associated with G and
X. Then we investigate some important properties of this vector space. Also, we
study some basic properties of the induced linear operators on the generalized
Cartesian symmetry Classes. Some open problems are also given.
