Definition
Let be a finite-dimensional representation of a group .
The character is the function
Representation determined by characters
Let be a reductive group. Then representations if and only if .
Assuming isomorphic representations
If and are isomorphic representations of , then .
Proof
Let be the isomorphism, then we have the diagram for
where is the (respective) action map on or .
Then
Assuming equal characters
Let and be representations of such that , then .
Proof
Since both and are reductive, we may decompose into direct sums of simple representations
Looking at the characters gives
since are all linearly independent (they form an orthogonal basis of the class functions for finite groups), then and the representations are isomorphic.
Properties