# Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree

## Statement

Suppose is a finite group, is a splitting field for (in particular, this includes any algebraically closed field of characteristic relatively prime to the order of ), and is the regular representation of over , i.e., the permutation representation corresponding to the regular group action.

Suppose are the irreducible representations of (up to equivalence of linear representations). Then, we have:

where is the degree of . In other words, the regular representation is the sum of all irreducible representations, with each irreducible representation occurring as many times as its degree.

## Related facts

- Peter-Weyl theorem
- Sum of squares of degrees of irreducible representations equals order of group
- Group ring over splitting field is direct sum of matrix rings for each irreducible representation

## 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 . is the regular representation of .

**To prove**:

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