Frobenius reciprocity

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This fact is related to: linear representation theory
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Statement

Plain statement

Let H \le G be finite groups, k be a field whose characteristic does not divide the order of G, and f_1,f_2 be class functions on H and G respectively taking values in k. In other words, f_1:H \to k and f_2:G \to k are functions with the property that f_1 is constant on conjugacy classes in H and f_2 is constant on conjugacy classes in G.

Then, the inner product of \operatorname{Ind}(f_1) with f_2 (in G) equals the inner product of f_1 and \operatorname{Res}(f_2) (in H).

In symbols:

\langle \operatorname{Ind}(f_1)_H^G,f_2 \rangle_G = \langle f_1, \operatorname{Res}(f_2)_H^G\rangle_H

This also applies in particular to the case when f_1,f_2 are the characters of linear representations.

Category-theoretic statement

Let H \le G.

Consider the category of representations of G (viz G-modules) and the category of representations of H, with a homomorphism of objects within a category, a vector space homomorphism that commutes with the G-action. Then, induction defines a functor from the category of H-modules to the category of G-modules, and restriction defines a functor from the category of G-modules to the category of H-modules. Frobenius reciprocity says that these two functors are adjoint functors.

Applications

Applications to relation between representation theory of group and subgroup