# Irreducible character of degree greater than one takes value zero on some conjugacy class

From Groupprops

## Statement

Suppose is a finite group and is the character of an irreducible linear representation of over , such that the degree of the representation (and hence, of ) is greater than one. Then, there exists an element (and hence, a Conjugacy class (?)) such that .

## Related facts

- Zero-or-scalar lemma
- Conjugacy class of more than average size has character value zero for some irreducible character

## Examples

Representation | Degree | Conjugacy class(es) where the character is zero | Group | Linear representation theory information |
---|---|---|---|---|

standard representation of symmetric group:S3 | 2 | -- class of 2-transpositions | symmetric group:S3 | linear representation theory of symmetric group:S3 |

faithful irreducible representation of dihedral group:D8 | 2 | all the conjugacy classes of elements outside the center. | dihedral group:D8 | linear representation theory of dihedral group:D8 |

faithful irreducible representation of quaternion group | 2 | all the conjugacy classes of elements outside the center. | quaternion group | linear representation theory of quaternion group |

standard representation of symmetric group:S4 | 3 | -- class of 3-cycles | symmetric group:S4 | linear representation theory of symmetric group:S4 |

## Facts used

- Character orthogonality theorem
- Sufficiently large implies splitting: In particular, this exhibits a splitting field of characteristic zero that is a finite cyclotomic extension of the rationals.
- Cauchy-Schwartz inequality
- Characters are algebraic integers

## Proof

**Given**: A finite group , the character of an irreducible representation of degree greater than of the group . is the identity element, is the degree, so .

**To prove**: There exists such that .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Fact (1) | Use Fact (1), and move the to the other side. | |||

2 | Step (1) | Pull out the term for the identity element. | |||

3 | , i.e., the representation has degree more than one | Step (2) | Rearrange Step (2), use . | ||

4 | Suppose is a finite degree cyclotomic extension of that is a splitting field for . Let be the Galois group of the field extension . Then, acts on the set of irreducible representations, with an automorphism acting by: |
Fact (2) | Fact (2) guarantees the existence of such a . | ||

5 | Fact (3) | Steps (3), (4) | |||

6 | Step (5) | [SHOW MORE] | |||

7 | for every | Steps (2), (4) | [SHOW MORE] | ||

8 | Step (7) | [SHOW MORE] | |||

9 | Steps (6), (8) | Step-combination direct | |||

10 | Each is an integer, so each is a nonnegative integer. | Fact (4) | Step (4) (definition of ) | [SHOW MORE] | |

11 | For some , | Steps (9), (10) | [SHOW MORE] | ||

12 | For some , | Step (11) | Follows from Step (11) and the observation that . |