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


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