Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics

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Abstract
Symmetric transverse traceless tensor harmonics of arbitrary rank are constructed on spheres S n 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 S n 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.