Peter?Weyl theorem

The Peter?Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Peter, in the setting of a compact Lie group, G. It generalises the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

Related Topics:
Harmonic analysis - Topological group - Compact - Hermann Weyl - Lie group - Regular representation - F. G. Frobenius - Issai Schur

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To state the theorem we need first the idea of the Hilbert space over G, L2(G); this makes sense because Haar measure exists on G. Calling it H, the group G has a unitary representation on H by acting on the left, or on the right. This implies a representation of G×G (via ρ((h,k))(g)=f(h-1gk)).

Related Topics:
Hilbert space - Haar measure - Representation

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This representation decomposes into the sum of ar{r}otimes r for each finite irreducible unitary representation of G where ar{r} is the dual representation. That is, there is a direct sum description of H with the indexation by all the classes (up to isomorphism) of irreducible unitary representations of G.

Related Topics:
Irreducible unitary representation - Dual representation - Direct sum

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This implies immediately the structure of H for the left or right representations of G, which comes out as a direct sum of each ρ as many times as its dimension (always finite).

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