Character Expansion Methods for $\mathrm{U}\mathrm{S}\mathrm{p}(2N)$, $\mathrm{S}\mathrm{O}(n)$, and $\mathrm{O}(n)$ using the Characters of the Symmetric Group

In theories with supersymmetry, we can calculate a special partition function, known as the superconformal index. In particular, for a gauge group of $\mathrm{U}(N)$ and particles belonging to the adjoint representation, there is a fast method known as the character expansion method, which uses the characters of the symmetric group. In this paper, we extend this method to theories of particles belonging to specific representations of the gauge groups: $\mathrm{U}\mathrm{S}\mathrm{p}(2N)$, $\mathrm{S}\mathrm{O}(n)$, and $\mathrm{O}(n)$. Furthermore, we propose a formula, which gives the large $N$ limit without using the characters.