This question was asked on MSE some time ago, here, but got no attention.

The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric functions of the eigenvalues of $U$ and can be written in terms of power sum symmetric functions, $s_\lambda(U)=\sum_\mu c_{\lambda\mu}p_\mu(U)$. The coefficients are the characters of the permutation group.

My question is how this translates for the orthogonal group. If the character of an irreducible representation, $Tr(R_\lambda(O))$, is written in terms of power sums, $\sum_\mu d_{\lambda\mu}p_\mu(O)$, what is known about the coefficients?

Same question for writing Schur functions in terms of orthogonal characters, and vice-versa.

References would be appreciated.