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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
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<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>
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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" />.
+
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  $  \pi $
 +
run through a family  $  \Sigma $
 +
of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group  $  G $(
 +
cf. [[Representation of a topological group|Representation of a topological group]]). Let  $  \mathop{\rm dim}  \pi $
 +
be the dimension of a representation  $  \pi $
 +
and let  $  u _ {ij} ^ {( \pi ) } $
 +
be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions
  
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.
+
$$
 +
\sqrt { \mathop{\rm dim}  \pi } u _ {ij} ^ {( \pi ) } \  ( \pi \in \Sigma )
 +
$$
  
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.
+
form an orthonormal basis in the space  $  L _ {2} (G) $
 +
of square-summable functions with respect to the Haar measure on  $  G $(
 +
the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on  $  G $,
 +
which coincides with the set of finite linear combinations of the functions  $  u _ {ij} ^ {( \pi ) } $,
 +
$  \pi \in \Sigma $,
 +
is uniformly dense in the space of all continuous complex-valued functions in  $  G $.
  
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" />.
+
If  $  G=T $
 +
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  $  G $
 +
is dense in the algebra of all continuous functions on  $  G $,
 +
constant on classes of conjugate elements. Another consequence is that for any element  $  a \in G $,
 +
$  a \neq e $,
 +
there is an irreducible continuous representation  $  \phi $
 +
of  $  G $
 +
such that  $  \phi (a) \neq e $;
 +
if, on the other hand,  $  G $
 +
is a compact Lie group, then  $  G $
 +
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 $  \phi $
 +
of a compact group $  G $
 +
in a Fréchet space $  E $.  
 +
Then the subspace of representation elements of $  E $
 +
is dense in $  E $.  
 +
Here an element $  v \in E $
 +
is called a representation, or spherical or almost-invariant, element if the orbit $  \phi (G)v $
 +
generates a finite-dimensional subspace in $  E $.  
 +
This is applicable in particular to the case where $  E $
 +
is the space of sections of a certain smoothness class of smooth vector $  G $-
 +
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 $  G $.
  
 
The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .
 
The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .
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====References====
 
====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>
 
<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====
 
====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.
+
A representation element is now usually called a $  G $-
 +
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]]].
+
The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on $  G $
 +
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 $  L _ {2} (G) $(
 +
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====
 
====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>
 
<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>

Latest revision as of 16:40, 31 March 2020


A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let $ \pi $ run through a family $ \Sigma $ of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group $ G $( cf. Representation of a topological group). Let $ \mathop{\rm dim} \pi $ be the dimension of a representation $ \pi $ and let $ u _ {ij} ^ {( \pi ) } $ be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

$$ \sqrt { \mathop{\rm dim} \pi } u _ {ij} ^ {( \pi ) } \ ( \pi \in \Sigma ) $$

form an orthonormal basis in the space $ L _ {2} (G) $ of square-summable functions with respect to the Haar measure on $ G $( the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on $ G $, which coincides with the set of finite linear combinations of the functions $ u _ {ij} ^ {( \pi ) } $, $ \pi \in \Sigma $, is uniformly dense in the space of all continuous complex-valued functions in $ G $.

If $ G=T $ 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 $ G $ is dense in the algebra of all continuous functions on $ G $, constant on classes of conjugate elements. Another consequence is that for any element $ a \in G $, $ a \neq e $, there is an irreducible continuous representation $ \phi $ of $ G $ such that $ \phi (a) \neq e $; if, on the other hand, $ G $ is a compact Lie group, then $ G $ has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion [5], [6]. Suppose one is given a continuous linear representation $ \phi $ of a compact group $ G $ in a Fréchet space $ E $. Then the subspace of representation elements of $ E $ is dense in $ E $. Here an element $ v \in E $ is called a representation, or spherical or almost-invariant, element if the orbit $ \phi (G)v $ generates a finite-dimensional subspace in $ E $. This is applicable in particular to the case where $ E $ is the space of sections of a certain smoothness class of smooth vector $ G $- 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 $ G $.

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

References

[1a] F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" Math. Ann. , 97 (1927) pp. 737–755
[1b] F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" , Gesammelte Abhandlungen H. Weyl , III : 73 , Springer (1968) pp. 58–75
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)
[4] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)
[5] R.S. Palais, T.E. Stewart, "The cohomology of differentiable transformation groups" Amer. J. Math. , 83 : 4 (1961) pp. 623–644
[6] G.D. Mostow, "Cohomology of topological groups and solvmanifolds" Ann. of Math. , 73 : 1 (1961) pp. 20–48

Comments

A representation element is now usually called a $ G $- finite element.

The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on $ G $ 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. [a1], Chapt. 7, §2. There exists a generalized Peter–Weyl theorem for unimodular Lie groups, cf. [a1], Chapt. 14, §2. The description of $ L _ {2} (G) $( 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. [a2].

References

[a1] A.O. Barut, R. Raçzka, "Theory of group representations and applications" , 1–2 , PWN (1977)
[a2] A. Wawrzyńczyk, "Group representations and special functions" , Reidel (1984) pp. Sect. 4.4
[a3] A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1988) pp. 17
How to Cite This Entry:
Peter-Weyl theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peter-Weyl_theorem&oldid=44951
This article was adapted from an original article by A.L. OnishchikA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article