Cartesian symmetry classes associated with certain subgroups of S_m

Let $V$ be an $n$-dimensional inner product space. Assume $G$ is a subgroup of the symmetric group of degree $m$, and $\lambda$ is an irreducible character of $G$. Consider the \emph{Cartesian symmetrizer} $C_{\lambda}$ on the Cartesian space $\times^{m}V$ defined by \[ C_{\lambda} = \frac{\lambda(1)}{|G|}\sum_{\tau\in G} \lambda(\tau) Q(\tau). \] The vector space $ V^{\lambda}(G) = C_{\lambda}(\times^{m}V) $ is called the Cartesian symmetry class associated with $G$ and $\lambda$. In this paper, we give a formula for the dimension of the cyclic subspace $V^{\lambda}_{ij}$. Then we discuss the problem existing an $O$-basis for the Cartesian symmetry class $V^{\lambda}(G)$. Also, we compute the dimension of the symmetry class $V^{\lambda}(G)$ when $G = \langle \sigma_{1} \sigma_{2} \cdots \sigma_{p} \rangle$ or $G = <\sigma_{1}><\sigma_{2}> \cdots <\sigma_{k}>$, where $\sigma_i$ are disjoint cycles in $S_{m}$. The dimensions are expressed in terms of the Ramanujan sum. Additionally, we provide a necessary and sufficient condition for the existence of an $O$-basis for Cartesian symmetry classes associated with the irreducible characters of the dihedral group $D_{2m}$. The dimensions of these classes are also computed.