Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics

Symmetric transverse traceless tensor harmonics of arbitrary rank are constructed on spheres Sn of dimensionality n≥3, and the associated eigenvalues of the Laplacian are computed. It is shown that these tensor harmonics span the space of symmetric transverse traceless tensors on Sn and are eigenfunctions of the quadratic Casimir operator of the group O(n+1). The dimensionalities of the eigenspaces of the Laplacian are computed for harmonics of rank 1 and rank 2.