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

Statement

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

Suppose $\varphi_1,\varphi_2,\dots,\varphi_r$ are the irreducible representations of $G$ (up to equivalence of linear representations). Then, we have:

$\rho = d_1\varphi_1 \oplus d_2\varphi_2 \oplus \dots d_r\varphi_r$

where $d_i$ is the degree of $\varphi_1$. In other words, the regular representation is the sum of all irreducible representations, with each irreducible representation occurring as many times as its degree.

Facts used

1. Maschke's averaging lemma, which we use to say that every representation is completely reducible.
2. 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 $G$ with irreducible representations having characters $\chi_1, \chi_2,\dots, \chi_r$ and degrees $d_1, d_2, \dots, d_r$. $\rho$ is the regular representation of $G$.

To prove: $\rho = d_1\varphi_1 \oplus d_2\varphi_2 \oplus \dots d_r\varphi_r$

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $\alpha$ takes the value $|G|$ at the identity element of $G$, and zero elsewhere. [SHOW MORE]
2 The inner product $\langle \alpha, \chi_i \rangle$ equals $d_i$ for all $i$. Step (1) [SHOW MORE]
3 $\alpha$ is the sum $\sum_{i=1}^r d_i\chi_i$ and $\rho$ is the sum $\sum_{i=1}^r d_i\varphi_i$ Facts (1),(2) $\chi_i$ are characters of (all the) irreducible representations. Step (2) [SHOW MORE]