The irreducible representations (and hence irreducible characters) for $S_n$ are in bijective correspondence with partitions of $n$. These functions are constant on the conjugacy classes of $S_n$, which are also in bijective correspondence with partitions of $n$.
Let $\lambda$ be a partition of $n$ and and let $\chi_{\lambda}$ be the associated character. Let $\mu$ be any partition of $n$ and let $C_{\mu}$ be the associated conjugacy class. The Frobenius formula is a simple formula that calculates $\chi_{\lambda} (C_{\mu})$
https://en.wikipedia.org/wiki/Frobenius_formula
I am in the process of carrying out many calculations using the frobenius formula for $S_4$. I'm currently working on when $\lambda=4$, that is, the partition of $4$ into the tuple $(4)$. is it then true that $\chi_{\lambda}(C_{\mu})$ will be $0$ for every $\mu$?
I think this because in the expansion of:
$\prod_{i<j}(x_i-x_j) \prod_j P_j(x_1,..x_k)^{i_j}$
It seems that $\prod_{i<j}(x_i-x_j)$ will always be $0$ because there is no combination of $i,j$ s.t. $i<j$ because $k=1$.
edit: Also, I was wondering if when I need to take the product of all partitions of $n$, if that means that $(3,1,1)$ is different than $(1,3,1)$ If somebody could review the Frobenius formula and let me know if I'm doing something wrong (for some reason I feel like I am) I would appreciate it.