On the natural representation of $S(\Omega)$ into $L^2(P(\Omega))$: Discrete harmonics and Fourier transform

Abstract

Let $\Omega$ denote a non-empty finite set. Let $S(\Omega)$ stand for the symmetric group on $\Omega$ and let us write $P(\Omega)$ for the power set of $\Omega$. Let $\rho: S(\Omega) \to U(L^2(P(\Omega)))$ be the left unitary representation of $S(\Omega)$ associated with its natural action on $P(\Omega)$. We consider the algebra consisting of those endomorphisms of $L^2(P(\Omega))$ which commute with the action of $\rho$. We find an attractive basis $B$ for this algebra. We obtain an expression, as a linear combination of $B$, for the product of any two elements of $B$. We obtain an expression, as a linear combination of $B$, for the adjoint of each element of $B$. It turns out the Fourier transform on $P(\Omega)$ is an element of our algebra; we give the matrix which represents this transform with respect to $B$.