On the natural representation of S(Ω) into L2(Ρ(Ω)):: Discrete harmonics and Fourier transform

[摘要] Let Q denote a nonempty finite set. Let S(Q) denote the symmetric group on Q and let P(Omega) denote the power set of Omega. Let rho : S(Omega) --> U(L-2 (P(Omega))) be the left unitary representation of S(Q) associated with its natural action on Y(Q). We consider the algebra consisting of those endomorphisms of L-2 (P(Omega)) which commute with the action of p. 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(Q) is an element of our algebra; we give the matrix which represents this transform with respect to B. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-10-01

[关键词] symmetric group;finite symmetric space;Johnson association scheme;discrete Laplacian operator;Hahn polynomials;finite Fourier transform;Krawtchouk polynomials;Terwilliger algebra