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A continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520601.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520604.png" /> are mappings of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520605.png" /> into two topological vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520608.png" />. This concept is especially fruitful in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i0520609.png" /> is a group or an algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206010.png" /> are representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206011.png" />. The set of intertwining operators forms the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206012.png" />, which is a subspace of the space of all continuous linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206018.png" /> are called disjoint representations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206019.png" /> contains an operator that defines an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206023.png" /> are equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206024.png" /> are locally convex spaces, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206026.png" /> are their adjoints, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206028.png" /> are the representations contragredient to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206030.png" />, respectively (cf. [[Contragredient representation|Contragredient representation]]), then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206031.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206032.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206035.png" /> are finite-dimensional or unitary representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206036.png" /> is irreducible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206037.png" /> admits a subrepresentation equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206038.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052060/i05206039.png" />. See also [[Intertwining number|Intertwining number]].
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A continuous linear operator  $  T:  E _ {1} \rightarrow E _ {2} $
 +
such that  $  T \pi _ {1} ( x) = \pi _ {2} ( x) T $,
 +
where  $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are mappings of a set $  X $
 +
into two topological vector spaces $  E _ {1} $
 +
and $  E _ {2} $
 +
and $  x \in X $.  
 +
This concept is especially fruitful in the case when $  X $
 +
is a group or an algebra and $  \pi _ {1} , \pi _ {2} $
 +
are representations of $  X $.  
 +
The set of intertwining operators forms the space $  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $,  
 +
which is a subspace of the space of all continuous linear mappings from $  E _ {1} $
 +
to $  E _ {2} $.  
 +
If $  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) = ( 0) $
 +
and $  \mathop{\rm Hom} ( \pi _ {2} , \pi _ {1} ) = ( 0) $,  
 +
then $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are called disjoint representations. If $  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $
 +
contains an operator that defines an isomorphism of $  E _ {1} $
 +
and $  E _ {2} $,  
 +
then $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are equivalent. If $  E _ {1} , E _ {2} $
 +
are locally convex spaces, if $  E _ {1}  ^ {*} $
 +
and $  E _ {2}  ^ {*} $
 +
are their adjoints, and if $  \pi _ {1}  ^ {*} $
 +
and $  \pi _ {2}  ^ {*} $
 +
are the representations contragredient to $  \pi _ {1} $
 +
and $  \pi _ {2} $,  
 +
respectively (cf. [[Contragredient representation|Contragredient representation]]), then for any $  T \in  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $,  
 +
the operator $  T  ^ {*} $
 +
is contained in $  \mathop{\rm Hom} ( \pi _ {2}  ^ {*} , \pi _ {1}  ^ {*} ) $.  
 +
If $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are finite-dimensional or unitary representations and $  \pi _ {1} $
 +
is irreducible, then $  \pi _ {2} $
 +
admits a subrepresentation equivalent to $  \pi _ {1} $
 +
if and only if $  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) \neq ( 0) $.  
 +
See also [[Intertwining number|Intertwining number]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Shtern,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Shtern,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


A continuous linear operator $ T: E _ {1} \rightarrow E _ {2} $ such that $ T \pi _ {1} ( x) = \pi _ {2} ( x) T $, where $ \pi _ {1} $ and $ \pi _ {2} $ are mappings of a set $ X $ into two topological vector spaces $ E _ {1} $ and $ E _ {2} $ and $ x \in X $. This concept is especially fruitful in the case when $ X $ is a group or an algebra and $ \pi _ {1} , \pi _ {2} $ are representations of $ X $. The set of intertwining operators forms the space $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $, which is a subspace of the space of all continuous linear mappings from $ E _ {1} $ to $ E _ {2} $. If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) = ( 0) $ and $ \mathop{\rm Hom} ( \pi _ {2} , \pi _ {1} ) = ( 0) $, then $ \pi _ {1} $ and $ \pi _ {2} $ are called disjoint representations. If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $ contains an operator that defines an isomorphism of $ E _ {1} $ and $ E _ {2} $, then $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent. If $ E _ {1} , E _ {2} $ are locally convex spaces, if $ E _ {1} ^ {*} $ and $ E _ {2} ^ {*} $ are their adjoints, and if $ \pi _ {1} ^ {*} $ and $ \pi _ {2} ^ {*} $ are the representations contragredient to $ \pi _ {1} $ and $ \pi _ {2} $, respectively (cf. Contragredient representation), then for any $ T \in \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $, the operator $ T ^ {*} $ is contained in $ \mathop{\rm Hom} ( \pi _ {2} ^ {*} , \pi _ {1} ^ {*} ) $. If $ \pi _ {1} $ and $ \pi _ {2} $ are finite-dimensional or unitary representations and $ \pi _ {1} $ is irreducible, then $ \pi _ {2} $ admits a subrepresentation equivalent to $ \pi _ {1} $ if and only if $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) \neq ( 0) $. See also Intertwining number.

References

[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2] A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)
How to Cite This Entry:
Intertwining operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_operator&oldid=47402
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article