In mathematics, 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 Fritz Peter, in the setting of a compact topological group G (Peter & Weyl 1927).
The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur.
The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions.
Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L2(G).
on G given as the composition where π : G → GL(V) is a finite-dimensional (continuous) group representation of G, and L is a linear functional on the vector space of endomorphisms of V (e.g. trace), which contains GL(V) as an open subset.
The set of matrix coefficients of G is dense in the space of continuous complex functions C(G) on G, equipped with the uniform norm.This first result resembles the Stone–Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization.
A corollary of this result is that the matrix coefficients of G are dense in L2(G).
Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous actions on Hilbert spaces.
Notice how this generalises the special case of the one-dimensional Hilbert space, where U(C) is just the circle group.)
Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.To state the third and final part of the theorem, there is a natural Hilbert space over G consisting of square-integrable functions,
; this makes sense because the Haar measure exists on G. The group G has a unitary representation ρ on
Roughly it asserts that the matrix coefficients for G, suitably renormalized, are an orthonormal basis of L2(G).
Thus, where Σ denotes the set of (isomorphism classes of) irreducible unitary representations of G, and the summation denotes the closure of the direct sum of the total spaces Eπ of the representations π.
Suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ.
The theorem now asserts that the set of functions is an orthonormal basis of
in G. The space of square-integrable class functions forms a closed subspace of
, defined by In the notation above, the character is the sum of the diagonal matrix coefficients: An important consequence of the preceding result is the following: This result plays an important part in Weyl's classification of the representations of a connected compact Lie group.
[1] A simple but helpful example is the case of the group of complex numbers of magnitude 1,
In this case, the irreducible representations are one-dimensional and given by There is then a single matrix coefficient for each representation, the function The last part of the Peter–Weyl theorem then asserts in this case that these functions form an orthonormal basis for
In this case, the theorem is simply a standard result from the theory of Fourier series.
in terms of matrix coefficients as a generalization of the theory of Fourier series.
The irreducible representations of SU(2), meanwhile, are labeled by a non-negative integer
and can be realized as the natural action of SU(2) on the space of homogeneous polynomials of degree
The key to verifying this claim is to compute that for any two complex numbers
consisting of matrix coefficients amounts to finding an orthonormal basis consisting of hyperspherical harmonics, which is a standard construction in analysis on spheres.
The Peter–Weyl theorem—specifically the assertion that the characters form an orthonormal basis for the space of square-integrable class functions—plays a key role in the classification of the irreducible representations of a connected compact Lie group.
For any finite-dimensional G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V).
It is closed, since G is compact, and a subgroup of the Lie group GL(V).
It follows by a theorem of Élie Cartan that the image of G is a Lie group also.