the representation on the regular representation is faithful

Question

I am reading the proof of the following proposition.

Proposition. As algebras, $\mathbb{C} G \cong \bigoplus \mathrm{End}(W_i),$ where $G$ is a finite group and $W_i$ are irreducible representation of $G$.

The proof goes as follows.

A representation $G \to \mathrm{Aut}(W_i)$ extends to a homomorphism $\mathbb{C}G \to \mathrm{End}(W_i)$. Thus we have $\phi: \mathbb{C} G \to \bigoplus \mathrm{End}(W_i)$.

I don't understand the next step where we want to show $\phi$ is injective.

The text book says that "This is injective since the representation on the regular representation is faithful".

What is the representation on the regular representation? And why is it faithful?

I appreciate any help.

Answer 1

The "representation on the regular representation" is probably just a typo for "the regular representation". This is the representation on the vector space $\mathbb CG$ in which each element of $G$ acts by multiplication from the left (on the standard basis vectors of $\mathbb CG$). That representation is faithful because every non-identity element of $G$ moves some vectors in $\mathbb CG$, for example any element of $G$ (considered as a basis vector in $\mathbb CG$). Finally, if we decompose the regular representation into irreducible summands, then each non-identity element must act nontrivially in at least one of those irreducible representations.