Let $x$ denote a set of $N$ variales. Schur functions $s_\lambda(x)$ and power sums $p_\lambda(x)$ are homogeneous symmetric polynomials indexed by partitions.
I believe the Schur function indexed by the partition of zero, $(0)\vdash 0$, must be set equal to unit, $s_0(x)=1$.
I believe the power sum indexed by the partition of zero equals the number of variables, $p_0(x)=\sum_{i=1}^N x_i^0=N$.
These two values do not seem to satisfy the relation $p_\mu(x)=\sum_\lambda \chi_\lambda(\mu)s_\lambda(x)$, which should lead to $p_0(x)=\chi_0(0)s_0(x)$. Here $\chi$ are irreducible characters of the permutation group.
I am note sure about the value of $\chi_0(0)$ (I would guess it is $1$), but in any case it seems to me that there is no value of this quantity that would fix this problem.
I know this is nitpicking, but it is bothering me.
You don't need to consider the power sum $p_0$ here. Instead you are considering the empty partition. $p_\lambda$ is defined as $\prod_{i=1}^l p_{\lambda_i}$, so if $\lambda$ is the empty partition then $p_\lambda$ is the empty product, so $p_\lambda = 1$, not $n$. Similarly, we would define $s_\lambda = 1$ if $\lambda$ is the empty partition. There is no group on $0$ elements, so I am not sure how to interpret its characters, but to satisfy the expression it seems reasonable to write $\chi_{\lambda}(\mu) = 1$ when $\lambda, \mu$ are empty partitions.