# Character determines representation in characteristic zero

## Statement

Suppose  is a finite group and  is a field of characteristic zero. Then, the character of any finite-dimensional representation of  over  completely determines the representation, i.e., no two inequivalent finite-dimensional representations can have the same character.

Note that  does not need to be a splitting field.

## Proof

Given: A group , two linear representations  of  with the same character  over a field  of characteristic zero.

To prove:  and  are equivalent as linear representations.

Proof: By Fact (2), both  and  are completely reducible, and are expressible as sums of irreducible representations. Suppose  is a collection of distinct irreducible representations obtained as the union of all the representations occurring in a decomposition of  into irreducible representations and a decomposition of  into irreducible representations. In other words, there are nonnegative integers  such that:



and



Let  denote the character of  and denote by  the value  (note: this would be 1 if  were a splitting field, and in general it is the sum of squares of multiplicities of irreducible constituents over a splitting field).

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1  and  for each  Fact (3) Direct application of fact
2  and  for each   has characteristic zero, so the manipulation makes sense Step (1)
3  for each   has characteristic zero Step (2) [SHOW MORE]
4  and  are equivalent Step (3)