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''intertwining number theorem.''
 
''intertwining number theorem.''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620701.png" /> be a finite group. The intertwining number between two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620703.png" />, is, by definition, the dimension of the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620704.png" />-homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620705.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620706.png" />.
+
Let $  G $
 +
be a finite group. The intertwining number between two representations $  \pi _ {i} : G \rightarrow  \mathop{\rm Aut} ( E _ {i} ) $,  
 +
$  i = 1 , 2 $,  
 +
is, by definition, the dimension of the space of $  G $-
 +
homomorphisms $  E _ {1} \rightarrow E _ {2} $:  
 +
$  i ( \pi _ {1} , \pi _ {2} ) = \mathop{\rm dim} (  \mathop{\rm Hom} _ {G} ( E _ {1} , E _ {2} ) ) $.
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620707.png" /> be subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620708.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m0620709.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207010.png" /> double coset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207011.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207012.png" /> is a set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207013.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207014.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207015.png" /> be a unitary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207016.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207017.png" /> be the corresponding [[Induced representation|induced representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207019.png" />. Consider the intertwining number between the unitary representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207021.png" /> of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207023.png" />. Then this number only depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207024.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207025.png" />). It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207026.png" />.
+
Now let $  H _ {1} , H _ {2} $
 +
be subgroups of $  G $,  
 +
and $  D $
 +
a $  ( H _ {2} , H _ {1} ) $
 +
double coset in $  G $(
 +
i.e. $  D $
 +
is a set of the form $  H _ {2} x H _ {1} $
 +
for some $  x \in G $).  
 +
Let $  \pi _ {i} $
 +
be a unitary representation of $  H _ {i} $
 +
and let $  \mathop{\rm Ind} _ {H _ {i}  }  ^ {G} ( \pi _ {i} ) $
 +
be the corresponding [[Induced representation|induced representation]] of $  G $,
 +
$  i = 1 , 2 $.  
 +
Consider the intertwining number between the unitary representations $  g \mapsto \pi _ {1} ( g) $
 +
and $  g \mapsto \pi _ {2} ( x g x  ^ {-} 1 ) $
 +
of the subgroup $  H _ {1} \cap x  ^ {-} 1 H _ {2} x $
 +
for some $  x \in D $.  
 +
Then this number only depends on $  D $(
 +
and $  \pi _ {1} , \pi _ {2} $).  
 +
It is denoted by $  i ( \pi _ {1} , \pi _ {2} , D ) $.
  
For the intertwining number between the induced representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207029.png" />, one now has the intertwining number formula
+
For the intertwining number between the induced representations $  \mathop{\rm Ind} _ {H _ {i}  }  ^ {G} ( \pi _ {i} ) $
 +
of $  G $,
 +
$  i = 1 , 2 $,  
 +
one now has the intertwining number formula
  
<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/m/m062/m062070/m06207030.png" /></td> </tr></table>
+
$$
 +
i (  \mathop{\rm Ind} _ {H _ {1}  }  ^ {G} ( \pi _ {1} ) ,\
 +
\mathop{\rm Ind} _ {H _ {2}  }  ^ {G} ( \pi _ {2} ) )  = \
 +
\sum _ { D } i ( \pi _ {1} , \pi _ {2} , D ) ,
 +
$$
  
where the sum is over the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207031.png" /> double cosets.
+
where the sum is over the set of all $  ( H _ {2} , H _ {1} ) $
 +
double cosets.
  
The Frobenius reciprocity theorem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207032.png" /> (cf. [[Induced representation|Induced representation]]) for representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207035.png" /> of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062070/m06207037.png" /> is an immediate consequence.
+
The Frobenius reciprocity theorem $  i ( \pi ,  \mathop{\rm Ind} _ {H}  ^ {G} ( \sigma ) ) = i (  \mathop{\rm Res} _ {H}  ^ {G} ( \pi ) , \sigma ) $(
 +
cf. [[Induced representation|Induced representation]]) for representations $  \pi $
 +
of $  G $
 +
and $  \sigma $
 +
of a subgroup $  H $
 +
of $  G $
 +
is an immediate consequence.
  
 
For a discussion of the intertwining number theorem for locally compact groups cf. [[#References|[a2]]].
 
For a discussion of the intertwining number theorem for locally compact groups cf. [[#References|[a2]]].

Latest revision as of 04:11, 6 June 2020


intertwining number theorem.

Let $ G $ be a finite group. The intertwining number between two representations $ \pi _ {i} : G \rightarrow \mathop{\rm Aut} ( E _ {i} ) $, $ i = 1 , 2 $, is, by definition, the dimension of the space of $ G $- homomorphisms $ E _ {1} \rightarrow E _ {2} $: $ i ( \pi _ {1} , \pi _ {2} ) = \mathop{\rm dim} ( \mathop{\rm Hom} _ {G} ( E _ {1} , E _ {2} ) ) $.

Now let $ H _ {1} , H _ {2} $ be subgroups of $ G $, and $ D $ a $ ( H _ {2} , H _ {1} ) $ double coset in $ G $( i.e. $ D $ is a set of the form $ H _ {2} x H _ {1} $ for some $ x \in G $). Let $ \pi _ {i} $ be a unitary representation of $ H _ {i} $ and let $ \mathop{\rm Ind} _ {H _ {i} } ^ {G} ( \pi _ {i} ) $ be the corresponding induced representation of $ G $, $ i = 1 , 2 $. Consider the intertwining number between the unitary representations $ g \mapsto \pi _ {1} ( g) $ and $ g \mapsto \pi _ {2} ( x g x ^ {-} 1 ) $ of the subgroup $ H _ {1} \cap x ^ {-} 1 H _ {2} x $ for some $ x \in D $. Then this number only depends on $ D $( and $ \pi _ {1} , \pi _ {2} $). It is denoted by $ i ( \pi _ {1} , \pi _ {2} , D ) $.

For the intertwining number between the induced representations $ \mathop{\rm Ind} _ {H _ {i} } ^ {G} ( \pi _ {i} ) $ of $ G $, $ i = 1 , 2 $, one now has the intertwining number formula

$$ i ( \mathop{\rm Ind} _ {H _ {1} } ^ {G} ( \pi _ {1} ) ,\ \mathop{\rm Ind} _ {H _ {2} } ^ {G} ( \pi _ {2} ) ) = \ \sum _ { D } i ( \pi _ {1} , \pi _ {2} , D ) , $$

where the sum is over the set of all $ ( H _ {2} , H _ {1} ) $ double cosets.

The Frobenius reciprocity theorem $ i ( \pi , \mathop{\rm Ind} _ {H} ^ {G} ( \sigma ) ) = i ( \mathop{\rm Res} _ {H} ^ {G} ( \pi ) , \sigma ) $( cf. Induced representation) for representations $ \pi $ of $ G $ and $ \sigma $ of a subgroup $ H $ of $ G $ is an immediate consequence.

For a discussion of the intertwining number theorem for locally compact groups cf. [a2].

References

[a1] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44
[a2] G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972) pp. Chapt. V
How to Cite This Entry:
Mackey intertwining number theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mackey_intertwining_number_theorem&oldid=47746