The dimensions of the symmetry classes of polynomials with respect to a certain cyclic subgroup of Sm generated by an m-cycle are explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed in terms of the Euler ϕ-function and the Möbius function for some special cases.

In this paper, we obtain the dimensions of symmetry classes of polynomials with respect to the irreducible characters of the dicyclic group as a subgroup of the full symmetric group. Then we discuss the existence of o-basis of these classes. In particular, the existence of o-basis of symmetry classes of polynomials with respect to the irreducible characters of the generalized quaternion group are concluded.

Let G be a subgroup of S m and suppose is an irreducible complex character of G. Let H d G be the symmetry class of polynomials of degree d with respect to G and . Let V be an d + 1 -dimensional inner product space over and V G be the symmetry class of tensors associated with G and . A monomorphism H d G → V G is given and it is used to obtain necessary and sufficient conditions for nonvanishing H d G . The nonexistence of o-basis of H d S m for a certain class of irreducible characters of S m is concluded. The dimensions of symmetry classes of polynomials with respect to the irreducible characters of S m and A m are computed.

The aim of this article is to study symmetry classes of tensors associated with Young subgroups of the symmetric group. We will discuss the dimension as well as *-bases of these types of symmetry classes.

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