Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree
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.
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.
- 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
- 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 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 .
|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]|