If a subgroup of
has a group representation
, then there is a unique
induced representation of
on a vector space
. The original space
is contained in
, and in fact,
(1)
|
where
is a copy of
.
The induced representation on
is denoted
.
Alternatively, the induced representation is the CG-module
(2)
|
Also, it can be viewed as -valued functions on
which commute with the
action.
(3)
|
The induced representation is also determined by its universal property:
(4)
|
where
is any representation of
. Also, the induced representation satisfies the following
formulas.
1. .
2.
for any group representation
.
3.
when
.
Some of the group characters of can be calculated from the group
characters of
, as induced representations, using Frobenius
reciprocity. Artin's reciprocity theorem
says that the induced representations of cyclic subgroups
of a finite group
generates a lattice of finite
index in the lattice of virtual characters.
Brauer's theorem says that the virtual characters
are generated by the induced representations from P-elementary
subgroups.