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

• Wait, the formula you give for $o_\lambda$ in your page as a determinant of a difference of $h$-functions is symmetric in all its arguments. I don't understand. – Marcel Jun 14 '19 at 17:14
• @Marcel : ah, i should add a warning - on the top of the page, it is stated what the alphabet is, (which includes the negative powers!). – Per Alexandersson Jun 14 '19 at 17:42
• Exactly, so the function $o_\lambda(x)$ has $2n$ arguments and has full $S_{2n}$ symmetry with respect to them, yes? So why did you suggest in your answer to treat the positive and negative alphabets separately? – Marcel Jun 14 '19 at 17:46
• @Marcel: Ah you are right of course! Hm, so there is a nice relationship using $\omega$ between orthogonal and symplectic Schur, so if there is a Murnaghan-Nakayama rule for one of these, then there is one for the other. – Per Alexandersson Jun 14 '19 at 19:05

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.