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

Abstract

Let $Ω$ denote a non-empty finite set. Let $S(Ω)$ stand for the symmetric group on $Ω$ and let us write $P(Ω)$ for the power set of $Ω$. Let $ρ: S(Ω) \to U(L^2(P(Ω)))$ be the left unitary representation of $S(Ω)$ associated with its natural action on $P(Ω)$. We consider the algebra consisting of those endomorphisms of $L^2(P(Ω))$ which commute with the action of $ρ$. 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 $P(Ω)$ is an element of our algebra; we give the matrix which represents this transform with respect to $B$.