character of group representation

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

todo - Lecture 29

Assuming equal characters

Let and be representations of such that , then .

Proof

todo - Lecture 30

Properties

• .