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 $λ$ is an irreducible character of $G$. Consider the \emph{Cartesian symmetrizer} $C_λ$ on the Cartesian space $\times^{m}V$ defined by \[ C_λ = \frac{λ(1)}{|G|}\sum_{τ\in G} λ(τ) Q(τ). \] The vector space $ V^λ(G) = C_λ(\times^{m}V) $ is called the Cartesian symmetry class associated with $G$ and $λ$. In this paper, we give a formula for the dimension of the cyclic subspace $V^λ_{ij}$. Then we discuss the problem existing an $O$-basis for the Cartesian symmetry class $V^λ(G)$. Also, we compute the dimension of the symmetry class $V^λ(G)$ when $G = \langle σ_{1} σ_{2} \cdots σ_{p} \rangle$ or $G = <σ_{1}><σ_{2}> \cdots <σ_{k}>$, where $σ_i$ 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.