Why is dimension of irrep equal to number of copies of its associated submodule? Let $G$ be a finite group. There exists a decomposition, as a $\mathbb{C}[G]$-module, $\mathbb{C}[G]=n_{1}V_{1}\oplus \cdots\oplus n_{r}V_{r}$, where the $V_{i}$ form a complete set of irreducible submodules of $\mathbb{C}[G]$, these submodules corresponding to the irreducible representations of $G$.
Why is the dimension of the representation $V_{i}$ equal to $n_{i}$, the number of copies of $V_{i}$ in the decomposition? I can prove that the space of $\mathbb{C}[G]$-module homomorphisms from $V_{1}$ to $V$ has dimension $n_{1}$, but I don't quite see how this implies that the dimension of the representation is also $n_{1}$.
 A: For any $\mathbb{C}[G]$-module $V$ there is a natural isomorphism of vector spaces
$$Hom_{\mathbb{C}[G]}(\mathbb{C}[G],V)\to V$$
given by $f\mapsto f(1)$ (the inverse of the isomorphism is $v\mapsto (\alpha\mapsto\alpha\cdot v)$). If $V$ is irreducible then, as you noticed, $\dim Hom_{\mathbb{C}[G]}(\mathbb{C}[G],V)$ is the multiplicity of $V$ in $\mathbb{C}[G]$, so this multiplicity is $\dim V$.
A: Another way to see this result is to note that $\mathbb{C}[G]$ naturally acts by vector space endomorphisms of its irreducible representations; that is, there is a canonical algebra homomorphism
$$\mathbb{C}[G] \to \bigoplus \text{End}(V_i).$$
Since $\mathbb{C}[G]$ is semisimple and has a faithful representation (namely itself), this map is injective. By the Jacobson density theorem, it is also surjective, hence bijective. In fact it is an isomorphism of $\mathbb{C}[G]\text{-}\mathbb{C}[G]$-bimodules, whence
$$\mathbb{C}[G] \cong \bigoplus V_i \otimes V_i^{\ast}$$
(where the first factor is acted on by the first copy of $\mathbb{C}[G]$ and the second factor is acted on by the second) and the conclusion follows. 
Of course, there is also the standard argument using characters, but I always found this result rather mysterious from the character-theoretic point of view.
