Mackey intertwining number theorem

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