View source for Mackey intertwining number theorem

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''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|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|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]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)  pp. §44</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Warner,   "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)  pp. Chapt. V</TD></TR></table>