# Characters of orthogonal groups as symmetric functions

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?

References would be appreciated.

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 $$p_{\lambda}(x_1,x_2,\dotsc,x_n)p_\mu(x_1^{-1},x_2^{-1},\dotsc,x_n^{-1})$$.