View source for Peter-Weyl theorem

A theorem on the approximation of functions on a compact [[Topological group|topological group]] by means of representation functions (cf. [[Representation function|Representation function]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724401.png" /> run through a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724402.png" /> of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724403.png" /> (cf. [[Representation of a topological group|Representation of a topological group]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724404.png" /> be the dimension of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724405.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724406.png" /> be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724407.png" /></td> </tr></table>

form an orthonormal basis in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724408.png" /> of square-summable functions with respect to the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p0724409.png" /> (the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244010.png" />, which coincides with the set of finite linear combinations of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244012.png" />, is uniformly dense in the space of all continuous complex-valued functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244013.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244014.png" /> is the rotation group for the plane, this assertion coincides with an elementary theorem on approximating periodic continuous functions by trigonometric polynomials.

A consequence of the Peter–Weyl theorem is that the set of linear combinations of characters of the irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244015.png" /> is dense in the algebra of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244016.png" />, constant on classes of conjugate elements. Another consequence is that for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244018.png" />, there is an irreducible continuous representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244021.png" />; if, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244022.png" /> is a compact Lie group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244023.png" /> has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion [[#References|[5]]], [[#References|[6]]]. Suppose one is given a continuous linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244024.png" /> of a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244025.png" /> in a Fréchet space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244026.png" />. Then the subspace of representation elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244027.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244028.png" />. Here an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244029.png" /> is called a representation, or spherical or almost-invariant, element if the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244030.png" /> generates a finite-dimensional subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244031.png" />. This is applicable in particular to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244032.png" /> is the space of sections of a certain smoothness class of smooth vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244034.png" />-fibrations, for example, the space of tensor fields of a certain type or given smoothness class on a smooth manifold with a smooth action of a compact Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244035.png" />.

The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe"  ''Math. Ann.'' , '''97'''  (1927)  pp. 737–755</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" , ''Gesammelte Abhandlungen H. Weyl'' , '''III''' :  73 , Springer  (1968)  pp. 58–75</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.S. Palais,   T.E. Stewart,   "The cohomology of differentiable transformation groups"  ''Amer. J. Math.'' , '''83''' :  4  (1961)  pp. 623–644</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.D. Mostow,   "Cohomology of topological groups and solvmanifolds"  ''Ann. of Math.'' , '''73''' :  1  (1961)  pp. 20–48</TD></TR></table>



====Comments====
A representation element is now usually called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244037.png" />-finite element.

The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244038.png" /> is known as the Weyl approximation theorem. The Peter–Weyl theorem gives a complete description of the (left or right) regular representation in terms of its irreducible components. In particular, each irreducible component occurs with a multiplicity equal to its dimension, cf. [[#References|[a1]]], Chapt. 7, §2. There exists a generalized Peter–Weyl theorem for unimodular Lie groups, cf. [[#References|[a1]]], Chapt. 14, §2. The description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072440/p07244039.png" /> (and the other unitary representations) in terms of the irreducible representations, including the fact that the irreducible unitary representations are finite dimensional, is known as Peter–Weyl theory, cf. e.g. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.O. Barut,   R. Raçzka,   "Theory of group representations and applications" , '''1–2''' , PWN  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Wawrzyńczyk,   "Group representations and special functions" , Reidel  (1984)  pp. Sect. 4.4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.W. Knapp,   "Representation theory of semisimple groups" , Princeton Univ. Press  (1988)  pp. 17</TD></TR></table>