On the Natural Representation of S(Ω) into L2(P(Ω)): Discrete Harmonics and Fourier Transform

Let Ω denote a nonempty finite set. Let S(Ω) denote the symmetric group on Ω and let P (Ω) denote the power set of Ω. Let ρ : S(Ω)→U(L2(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 L2(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 that the Fourier transform on P(Ω) is an element of our algebra; we give the matrix which represents this transform with respect to B.