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The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520501.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520502.png" /> of intertwining operators (cf. [[Intertwining operator|Intertwining operator]]) for two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520504.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520505.png" /> into topological vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520507.png" />, respectively. The concept of the intertwining number is especially fruitful in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520508.png" /> is a group or an algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i0520509.png" /> are representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205010.png" />. Even for finite-dimensional representations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205011.png" /> in general, but for finite-dimensional representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205014.png" /> the following relations hold:
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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/i/i052/i052050/i05205015.png" /></td> </tr></table>
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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/i/i052/i052050/i05205016.png" /></td> </tr></table>
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The dimension  $  c ( \pi _ {1} , \pi _ {2} ) $
 +
of the space  $  \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $
 +
of intertwining operators (cf. [[Intertwining operator|Intertwining operator]]) for two mappings  $  \pi _ {1} $
 +
and  $  \pi _ {2} $
 +
of a set  $  X $
 +
into topological vector spaces  $  E _ {1} $
 +
and  $  E _ {2} $,
 +
respectively. The concept of the intertwining number is especially fruitful in the case when  $  X $
 +
is a group or an algebra and  $  \pi _ {1} , \pi _ {2} $
 +
are representations of  $  X $.  
 +
Even for finite-dimensional representations,  $  c ( \pi _ {1} , \pi _ {2} ) \neq c ( \pi _ {2} , \pi _ {1} ) $
 +
in general, but for finite-dimensional representations  $  \pi _ {1} $,
 +
$  \pi _ {2} $,
 +
$  \pi _ {3} $
 +
the following relations hold:
  
while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205017.png" /> is a group, then also
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$$
 +
c ( \pi _ {1} \oplus \pi _ {2} , \pi _ {3} )  = \
 +
c ( \pi _ {1} , \pi _ {3} ) + c ( \pi _ {2} , \pi _ {3} );
 +
$$
  
<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/i/i052/i052050/i05205018.png" /></td> </tr></table>
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$$
 +
c ( \pi _ {1} , \pi _ {2} \oplus \pi _ {3} )  = c ( \pi _ {1} , \pi _ {2} ) + c ( \pi _ {1} , \pi _ {3} ),
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205020.png" /> are irreducible and finite dimensional or unitary, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205021.png" /> is equal to 1 or 0, depending on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052050/i05205023.png" /> are equivalent or not. For continuous finite-dimensional representations of a compact group, the intertwining number can be expressed in terms of the characters of the representations (cf. also [[Character of a representation of a group|Character of a representation of a group]]).
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while if  $  X $
 +
is a group, then also
 +
 
 +
$$
 +
c ( \pi _ {1} \otimes \pi _ {2} , \pi _ {3} )  = \
 +
c ( \pi _ {1} , \pi _ {2}  ^ {*} \otimes \pi _ {3} ).
 +
$$
 +
 
 +
If  $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are irreducible and finite dimensional or unitary, then $  c ( \pi _ {1} , \pi _ {2} ) $
 +
is equal to 1 or 0, depending on whether $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are equivalent or not. For continuous finite-dimensional representations of a compact group, the intertwining number can be expressed in terms of the characters of the representations (cf. also [[Character of a representation of a group|Character of a representation of a group]]).
  
 
====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


The dimension $ c ( \pi _ {1} , \pi _ {2} ) $ of the space $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $ of intertwining operators (cf. Intertwining operator) for two mappings $ \pi _ {1} $ and $ \pi _ {2} $ of a set $ X $ into topological vector spaces $ E _ {1} $ and $ E _ {2} $, respectively. The concept of the intertwining number is especially fruitful in the case when $ X $ is a group or an algebra and $ \pi _ {1} , \pi _ {2} $ are representations of $ X $. Even for finite-dimensional representations, $ c ( \pi _ {1} , \pi _ {2} ) \neq c ( \pi _ {2} , \pi _ {1} ) $ in general, but for finite-dimensional representations $ \pi _ {1} $, $ \pi _ {2} $, $ \pi _ {3} $ the following relations hold:

$$ c ( \pi _ {1} \oplus \pi _ {2} , \pi _ {3} ) = \ c ( \pi _ {1} , \pi _ {3} ) + c ( \pi _ {2} , \pi _ {3} ); $$

$$ c ( \pi _ {1} , \pi _ {2} \oplus \pi _ {3} ) = c ( \pi _ {1} , \pi _ {2} ) + c ( \pi _ {1} , \pi _ {3} ), $$

while if $ X $ is a group, then also

$$ c ( \pi _ {1} \otimes \pi _ {2} , \pi _ {3} ) = \ c ( \pi _ {1} , \pi _ {2} ^ {*} \otimes \pi _ {3} ). $$

If $ \pi _ {1} $ and $ \pi _ {2} $ are irreducible and finite dimensional or unitary, then $ c ( \pi _ {1} , \pi _ {2} ) $ is equal to 1 or 0, depending on whether $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent or not. For continuous finite-dimensional representations of a compact group, the intertwining number can be expressed in terms of the characters of the representations (cf. also Character of a representation of a group).

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 number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=47401
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article