The Reynolds operator (or averaging operator) associated with a group representation 
of a finite group 
on a finite-dimensional complex
vector space 
is
 |
(1)
|
It is the projection operator onto the fixed subspace 
. Indeed, for every 
,
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Taking the matrix trace gives the character formula
 |
(5)
|
where 
denotes the matrix trace and 
is the group character
of 
.
In invariant theory, the same average sends a polynomial to its invariant part and is also called the Reynolds operator (Sturmfels 2008).
