## Frobenius Reciprocity

**Point of Post: **In this post we discuss the Frobenius Reciprocity Theorem and discuss some of its consequences including its obvious relation to multiplicities.

*Motivation*

In our last post we saw that given a finite group and some there is a natural map by extending linearly the map for every where is the induced character. We saw dually there was a map which just took a class function on and restricted it to . In this post we prove the amazing fact that the maps and are adjoint, in the sense that . What of course then shall be true is that the maps and are adjoint. From this we shall derive a fascinating result about the relation between how often an irrep occurs in the decomposition of and how often it occurs in .

*Frobenius Reciprocity Theorem*

We waste no time and proceed with the theorem. The most beautiful thing about this theorem though is how simple the proof is–it’s literally just a simple computation, but it is very, very powerful.

**Theorem (Frobenius Reciprocity): ***Let be a finite group and . Then, the two maps and are adjoint. In other words, for any and *

**Proof: **Let and be arbitrary and we recall that if we fix some transveral for then

where is zero on and on . Then, by definition one has

where we used the obvious fact that the only places where is precisely and summing over this is precisely summing over where we replace by –and the fact that is a class function on so that for every .

Note that if we were doing character theory and solely interested in the characters of and not the representations then we could have *defined *the induced map to be equal to the unique adjoint of and then defined the induced character of a character in to be the fact that it would be character follows from the following:

*Multiplicities of Irreps*

We now discuss a fantastic corollary of the Frobenius Reciprocity theorem. The beginnings of which come from the offhand remark at the end of the above section. Namely that it somehow follows from the Frobenius Reciprocity theorem that is a character of whenever is a character of . We’ll recall that this reduces to showing that or that for every . But, it’s trivial that is a character of (it’s the character of the restriction of to if is the character for ) and so from where the rest follows from Frobenius Reciprocity theorem. We’d now like to extend this observation to show a brilliant fact about the multiplicities of an induced representation. Namely, for a group and a representation on we define, for each , the *multiplicity of in *denoted to be the number of times an element of occurs in the decomposition of into irreps. But, it’s trivial using the orthogonality of irreducible characters to note that . The brilliant thing though is that given two irreps of and of we see that

Which, when you think about it, is a crazy thing. The number of times something in occurs in the decomposition of an induced representation of something in is equal to the number of times something in appears in the decomposition of the restriction. Just amazing.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Math. Soc., 1996. Print.

[…] and are the same representation. In general this may be a very messy task, but thanks to the Frobenius Reciprocity theorem this becomes almost obvious in the sense that since is the adjoint of it suffices to show that […]

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[…] of . The idea is simple, namely we know that being an irrep is equivalent to having but from Frobenius Reciprocity theorem we know that this is equivalent to showing the slightly more scary looking . But, thanks to […]

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