This question was asked on MSE some time ago, [here](https://math.stackexchange.com/questions/3128810/characters-of-orthogonal-groups-as-symmetric-functions), 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.



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](https://www.math.upenn.edu/~peal/polynomials/schurMisc.htm#schurOrthogonal)
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 $p\_{\lambda}(x\_1,x\_2,\dotsc,x\_n)p\_\mu(x\_1^{-1},x\_2^{-1},\dotsc,x\_n^{-1})$.


