# Sum of squares of degrees of irreducible representations equals order of group

This fact is related to: linear representation theory

View other facts related to linear representation theoryView terms related to linear representation theory |

## Contents

## Statement

Suppose is a finite group, is a splitting field for , and are the characters of the irreducible linear representations (up to equivalence) of over . Let be the degree of . In other words, are the Degrees of irreducible representations (?) of . Then:

This fact is instrumental in defining the Plancherel measure on the set of irreducible representations of a finite group, which assigns a measure of to the irreducible representation.

Note also that the s (up to rearrangement) are the same for all splitting fields -- see degrees of irreducible representations are the same for all splitting fields.

## Related facts

### Alternative formulations

- Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree
- Group ring over splitting field is direct sum of matrix rings for each irreducible representation
- Peter-Weyl theorem: A generalization to compact groups.

### Similar facts

- Number of irreducible representations equals number of conjugacy classes
- Character orthogonality theorem
- Column orthogonality theorem
- Grand orthogonality theorem

See also the many facts about the degrees of irreducible representations under degrees of irreducible representations.

## Particular cases

### Groups

Group/group type | Order | Degrees of irreducible representations | Squares of degrees of irreducible representations |
---|---|---|---|

trivial group | 1 | 1 | 1 |

abelian group of order | 1 ( times) | 1 ( times) | |

symmetric group:S3 | 6 | 1,1,2 | 1,1,4 |

dihedral group:D8 | 8 | 1,1,1,1,2 | 1,1,1,1,4 |

quaternion group | 8 | 1,1,1,1,2 | 1,1,1,1,4 |

dihedral group:D10 | 10 | 1,1,2,2 | 1,1,4,4 |

alternating group:A4 | 12 | 1,1,1,3 | 1,1,1,9 |

direct product of S3 and Z2 | 12 | 1,1,1,1,2,2 | 1,1,1,1,4,4 |

dicyclic group:Dic12 | 12 | 1,1,1,1,2,2 | 1,1,1,1,4,4 |

symmetric group:S4 | 24 | 1,1,2,3,3 | 1,1,4,9,9 |

special linear group:SL(2,3) | 24 | 1,1,1,2,2,2,3 | 1,1,1,4,4,4,9 |

### Families

Family of groups | Order | Degrees of irreducible representations | Family-specific explanation for sum of squares |
---|---|---|---|

dihedral group of even degree | ( times), ( times) | algebraic simplification. See also linear representation theory of dihedral groups | |

dihedral group of odd degree | ( times), ( times) | algebraic simplification. See also linear representation theory of dihedral groups | |

symmetric group of degree | For each partition of , an irreducible representation of degree , which is the number of Young tableaux of shape | Robinson-Schensted correspondence. See also linear representation theory of symmetric groups | |

general affine group of degree one over a finite field of size | ( times), ( time) | See linear representation theory of general affine group of degree one over a finite field | |

general linear group of degree two over a finite field of size | 1 ( times), ( times), ( times), ( times) | Algebraic simplification. See linear representation theory of general linear group of degree two over a finite field | |

projective general linear group of degree two over a finite field of size | Case odd: 1 (2 times), ( times), (2 times), ( times) Case even: 1 (1 time), ( times), (1 time), ( times) |
Algebraic simplification. See linear representation theory of projective general linear group of degree two over a finite field | |

special linear group of degree two over a finite field of size | Case odd: 1 (1 time), (2 times), (2 times), ( times), (1 time), ( times) Case even: 1 (1 time), ( times), (1 time), ( times) |
Algebraic simplification. See linear representation theory of special linear group of degree two over a finite field | |

projective special linear group of degree two over a finite field of size | if odd; if even | Case odd: 1 (1 time), (2 times), ( or times depending on congruence class of mod ), (1 time), ( or times depending on congruence class of mod ) Case even: 1 (1 time), ( times), (1 time), ( times) |
Algebraic simplification. See linear representation theory of projective special linear group of degree two over a finite field |

## Facts used

- Maschke's averaging lemma, which we use to say that every representation is completely reducible.
- Orthogonal projection formula, which in turn uses character orthogonality theorem. See inner product of functions for the notation.

## Proof

### Proof in characteristic zero

*Note*: We can in fact use this proof to also show that there are only finitely many equivalence classes of irreducible representations, though the formulation below does not quite do that.

**Given**: A finite group with irreducible representations having characters and degrees .

**To prove**:

**Proof**: We let be the regular representation of , i.e., the permutation representation obtained by using the regular group action. Let be the character of .

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

1 | takes the value at the identity element of , and zero elsewhere. | [SHOW MORE] | |||

2 | The inner product equals for all . | Step (1) | [SHOW MORE] | ||

3 | is the sum | Facts (1),(2) | are characters of (all the) irreducible representations. | Step (2) | [SHOW MORE] |

4 | The value of at the identity element is . | Step (3) | [SHOW MORE] | ||

5 | Steps (1), (4) | [SHOW MORE] |

### Proof in other characteristics

This follows from the characteristic zero proof, and the fact that degrees of irreducible representations are the same for all splitting fields.