Cartesian symmetry classes associated with certain subgroups of S_m

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 S_m (generated by a product of disjoint cycles) and the product of cyclic subgroups of S_m 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.

Cartesian symmetry classes associated with certain subgroups of S_m Keywords:

Irreducible characters , Cartesian symmetry classes , orthogonal basis , cyclic group , Dihedral group , Ramanujan sum