Difference between revisions of "Mackey intertwining number theorem"
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''intertwining number theorem.'' | ''intertwining number theorem.'' | ||
| − | Let | + | 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 | + | 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 | + | 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 | + | where the sum is over the set of all $ ( H _ {2} , H _ {1} ) $ |
| + | double cosets. | ||
| − | The Frobenius reciprocity theorem | + | 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=17864
Mackey intertwining number theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mackey_intertwining_number_theorem&oldid=17864