An induced representation of a group 
is the group representation
on 
constructed from a group representation 
of a subgroup 
. Its underlying vector
space is 
.
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 
-module
 |
(2)
|
Also, it can be viewed as 
-valued functions on 
that commute with the 
action and have finite support modulo 
.
 |
(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 induction theorem
(tom Dieck 2009, §4.5) says that the induced representations of cyclic
subgroups of a finite group 
generate 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.
