Frobenius reciprocity
This fact is related to: linear representation theory
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Contents
Statement
Plain statement
Let be finite groups,
be a field whose characteristic does not divide the order of
, and
be class functions on
and
respectively taking values in
. In other words,
and
are functions with the property that
is constant on conjugacy classes in
and
is constant on conjugacy classes in
.
Then, the inner product of with
(in
) equals the inner product of
and
(in
).
In symbols:
This also applies in particular to the case when are the characters of linear representations.
Category-theoretic statement
Let .
Consider the category of representations of (viz
-modules) and the category of representations of
, with a homomorphism of objects within a category, a vector space homomorphism that commutes with the
-action. Then, induction defines a functor from the category of
-modules to the category of
-modules, and restriction defines a functor from the category of
-modules to the category of
-modules. Frobenius reciprocity says that these two functors are adjoint functors.
Applications
Applications to relation between representation theory of group and subgroup
- Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
- Maximum degree of irreducible representation of subgroup is less than or equal to maximum degree of irreducible representation of whole group