# Some irreducible character vanishes on every conjugacy class

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This fact is related to: linear representation theory

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## Contents

## Statement

Let be a finite group, and be a field of characteristic zero. Suppose is a conjugacy class in .

Then, there exists an irreducible linear representation of whose character takes the value zero on the conjugacy class.

## Proof

### Proof over the complex numbers

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### Proof over arbitrary fields of characteristic zero

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