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

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:100

Article

Marco, JM ; Parcet, J

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

DOI  :  10.1006/jcta.2002.3291

来源: Elsevier

【 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).

【 授权许可】

Free