Upper block triangular form for the Laplace-Beltrami operator on the special orthogonal group acquired through a flag of trace polynomials spaces

The Laplacian of a general trace polynomial function defined on the special orthogonal group $SO(N)$ is explicitly computed. An invariant flag of spaces generated by trace polynomials is constructed. The matrix of the Laplace-Beltrami operator on $SO(N)$ for this flag of vector spaces takes an upper block triangular form. As a consequence of this construction, the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on $SO(N)$ can be computed in an iterative manner. For the particular cases of the special orthogonal groups $SO(3)$ and $SO(4)$ the complete list of eigenvalues is obtained and the corresponding irreducible characters of the representation for these groups are expressed as trace polynomials.