Some irreducible character vanishes on every conjugacy class

From Groupprops
Jump to: navigation, search
This fact is related to: linear representation theory
View other facts related to linear representation theoryView terms related to linear representation theory |

Statement

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

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

Proof

Proof over the complex numbers

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Proof over arbitrary fields of characteristic zero

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]