On the Natural Representation of S(Ω) into L2(P(Ω)): Discrete Harmonics and Fourier Transform
✍ Scribed by José Manuel Marco; Javier Parcet
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 232 KB
- Volume
- 100
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
Let O denote a nonempty finite set. Let SðOÞ denote the symmetric group on O and let PðOÞ denote the power set of O: Let r : SðOÞ ! UðL 2 ðPðOÞÞÞ be the left unitary representation of SðOÞ associated with its natural action on PðOÞ: We consider the algebra consisting of those endomorphisms of L 2 ðPðOÞÞ which commute with the action of r: 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 PðOÞ is an element of our algebra; we give the matrix which represents this transform with respect to B: # 2002 Elsevier Science (USA)