Characters of orthogonal groups as symmetric functions

I might be totally wrong, but the analogues of Schur functions in the orthogonal case, the so called orthogonal characters are not polynomials in just the $x_i$, but polynomials in $x_i^{\pm 1}$. You can perhaps treat the negative alphabet separately, and expand in say $p_{\lambda}(x_1,x_2,\dotsc,x_n)p_\mu(x_1^{-1},x_2^{-1},\dotsc,x_n^{-1})$.

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.