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

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• 作者：Manuel Marco ; José ; Parcet ; Javier
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2002
• 出版时间：October, 2002
• 年：2002
• 卷：100
• 期：1
• 页码：153-175
• 全文大小：232 K

文摘

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.