At the moment I am studying random walks on the symmetric group $S_n$, and have stumbled across the following problem, which seems to be concrete enough to might have a known answer, even though I cannot find one.
My problem is the following; I am studying the space of all functions on the symmetric group into $ \mathbb{R} $, and it is quite easy to find a set of functions whose span is this space, e.g. the functions $ \psi_{i,j}(\sigma) = 1_ {\sigma(i) = j}$. However, I would want to have such a set of functions which is orthogonal with respect to the inner product $\langle \psi_1, \psi_2 \rangle = \mathbb{E}[\psi_1(\sigma) \psi_2(\sigma)] $ where the underlying probability measure is the uniform measure on $S_n$, i.e. a basis for the space of all functions on $S_n$, given my inner product. Is any such baseis, where the base vectors can be explicitly described, known?