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Mackey intertwining number theorem

From Encyclopedia of Mathematics
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intertwining number theorem.

Let be a finite group. The intertwining number between two representations , , is, by definition, the dimension of the space of -homomorphisms : .

Now let be subgroups of , and a double coset in (i.e. is a set of the form for some ). Let be a unitary representation of and let be the corresponding induced representation of , . Consider the intertwining number between the unitary representations and of the subgroup for some . Then this number only depends on (and ). It is denoted by .

For the intertwining number between the induced representations of , , one now has the intertwining number formula

where the sum is over the set of all double cosets.

The Frobenius reciprocity theorem (cf. Induced representation) for representations of and of a subgroup of 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