cartesian symmetry classes associated with certain subgroups of sm

In this paper, the problem existing o-basis for cartesian symmetry classes is discussed. the dimensions of cartesian symmetry classes associated with a cyclic subgroup of the symmetric group sm (generated by a product of disjoint cycles) and the product of cyclic subgroups of sm are explicitly expressed in terms of the ramanajun sum. additionally, a necessary and sufficient condition for the existence of an o-basis for cartesian symmetry classes associated with the irreducible characters of dihedral group is given. the dimensions of these classes are also computed.