Fourier analysis of subgroup conjugacy invariant functions on finite groups. (English) Zbl 1271.43006
The article deals with a variant of the Fourier transform on the convolution algebra \(\mathcal{C}(G,H)\) of complex-valued functions on a finite group \(G\), invariant under the conjugation action of its subgroup \(H\). An orthogonal basis of \(\mathcal{C}(G,H)\) is found and used to establish an algebra isomorphism – a “Fourier transform” – between \(\mathcal{C}(G,H)\) and a certain direct sum of matrix algebras. It is accompanied by a corresponding inversion formula and Plancherel’s formula. The theory is then specialized to the center of \(\mathcal{C}(G,H)\), yielding a spherical Fourier transform.
The authors also provide an example involving the symmetric group, and propose, under some additional assumptions, a canonical choice of the basis of \(\mathcal{C}(G,H)\).
The authors also provide an example involving the symmetric group, and propose, under some additional assumptions, a canonical choice of the basis of \(\mathcal{C}(G,H)\).
Reviewer: Łukasz Garncarek (Wrocław)
MSC:
| 43A90 | Harmonic analysis and spherical functions |
| 20C15 | Ordinary representations and characters |
| 20C30 | Representations of finite symmetric groups |
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