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.

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

 * 1) uses::Maschke's averaging lemma, which we use to say that every representation is completely reducible.
 * 2) uses::Orthogonal projection formula, which in turn uses 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 $$G$$ 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: $$\rho = d_1\varphi_1 \oplus d_2\varphi_2 \oplus \dots d_r\varphi_r$$