# Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials

The irreducible characters of the orthogonal group $$O(2N)$$ are given by $$o_\lambda(x_1,x_1^{-1},...x_N,x_N^{-1})=\frac{\det(x_j^{N+\lambda_i-i}+x_j^{-(N+\lambda_i-i)})}{\det(x_j^{N-i}+x_j^{-(N-i)})}$$

I was playing with them as basis for the space of homogeneous symmetric polynomials. I wanted to write the function $$p_3=\sum_{i=1}^N (x_i^3+x_i^{-3})$$ as a linear combination of $$o_\lambda$$'s.

I started with $$N=2$$ and in that case I found that $$p_3=o_{(3)}-2o_{(2,1)}+o_{(1)}.$$

However, I then tried with $$N=4$$ and in that case I found that $$p_3=o_{(3)}-o_{(2,1)}+o_{(1,1,1)}.$$

I was expecting the coefficients to be the same, i.e. to be independent of $$N$$. (Independence of $$N$$ indeed holds when using characters of the unitary group, in which case the coefficients are the characters of the permutation group.)

Have I made some mistake or are the coefficients indeed dependent on $$N$$?

The coefficients do depend on $$N$$. A way to get around this and deal with "universal characters" was found by Koike and Terada (Young-diagrammatic methods for the representation theory of the classical groups of type $$B_n$$, $$C_n$$, $$D_n$$).