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 $$C_{\lambda }=\frac{\lambda (1)}{\mathrm{\mid }G\mathrm{\mid }}\sum _{\tau \in G}\lambda (\tau )Q(\tau )\mathrm{.}$$ 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 $V_{ij}^{\lambda }$. 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 $G=⟨{\sigma }_1{\sigma }_2\cdots {\sigma }_p⟩$ 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.