How to calculate the Fourier Transform of a constant function on the symmetric group? Let $f: \mathcal{S}_n \to \mathbb{C}$. We define the Fourier Transform of $f$ as the collection of matrices
$$ \hat{f}_{\lambda} = \sum_{\sigma \in \mathcal{S}_n} f(\sigma) \rho_{\lambda}(\sigma)$$
indexed by the integer partitions of $n$, $\lambda$, where $\rho_{\lambda}$ is the irreducible representation of $\mathcal{S}_n$ corresponding to $\lambda$. For the purpose of this question, our choice of map from $\lambda$ to $\rho_{\lambda}$ doesn't matter, and our choice of representation matrices shouldn't matter.
I am trying to compute the Fourier transform of a constant function $f(\sigma)=a$ for some $a\in \mathbb{C}$ directly from the definition.
I believe I have found the Fourier transform through trial and error. If $\rho_{\lambda}$ is the trivial representation, then
$$ \hat{f}_{\lambda} = \sum_{\sigma \in \mathcal{S}_n} a \cdot [1] = [n!\cdot a]\text{.}$$
and if $\rho_{\lambda}$ is any other representation, then $\hat{f}_{\lambda}$ vanishes. This can be checked by plugging this guess for $\hat{f}$ into the inverse Fourier Transform.
I am particularly interested in the case where $\rho_{\lambda}(\sigma)$ are Young's Orthogonal Representation matrices. A note I found online indicated that the result should follow directly from the unitary property of these matrix representations.
 A: So first we may assume that $a=1$, since it can be pulled out of the definition. Thus you are asking what the sum
$$\alpha_\rho=\sum_{g\in G} \rho(g)$$
is for some representation $\rho$ of a finite group $G$. I assume that $\rho$ is irreducible to begin with, the general case follows from this.
If $\rho$ is trivial then $\alpha_\rho=|G|\cdot 1$, as you state. Assume that $\rho$ is non-trivial. If $g\in G$ then notice that $\alpha_\rho \rho(g)=\alpha_\rho$. In particular, this means that $\alpha_\rho$ commutes with $\rho$, hence is a scalar matrix $\lambda\cdot I$ by Schur's lemma. We only need to determine which scalar. So take traces. The left-hand side is $n\lambda$, where $n$ is the dimension of the representation $\rho$. The right-hand side is $$\sum_{g\in G}\chi(g),$$
where $\chi$ is the character afforded by $\rho$. This sum is equal to $\langle \chi,1_G\rangle$, which is zero as $\rho$ is non-trivial. Thus $\alpha_\rho=0$ if $\rho$ is non-trivial.
For the general case, $\rho$ is a sum of non-trivial irreducible representations---and hence $\alpha_\rho$ is a block-zero matrix, i.e., the zero matrix---and some number $m$ of trivial representations, and each contributes a $|G|$ somewhere on the diagonal.
