TITLE: Characters of orthogonal groups as symmetric functions
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: I don't know about writing ${\rm Tr}(R_\lambda(O))$ in terms of power sums, but the reverse procedure can be carried out using the so-called "characters of the Brauer algebra" (they are not really characters).
This theory is developed by Arun Ram in two papers:

"Characters of Brauer's centralizer algebra", Pacific J. Math. 169, p.173, 1995
"A ‘Second Orthogonality Relation’ for Characters of Brauer Algebras", Europ . J . Combinatorics 18, p. 685, 1997

In those works he gives some combinatorial way to compute such characters and also provides a formula for them in terms of the characters of the permutation group.