Explicit decomposition of the regular representation of a finite group It is known that if $G$ is a finite group then the regular representation $\mathbf{C}[G]$ decomposes as
$$
\mathbf{C}[G] = \bigoplus_{\pi} \pi^{\text{dim}(\pi)}
$$
where $\pi$ ranges over all irreducible representations of $\pi$.
My question is : is there an explicit way to understand this isomorphism? In particular if I give you such a $\pi$  can you then tell me how I can explicitely embed $\pi$ in $\mathbf{C}[G]$
When $\pi$ is of dimension $1$ it is easy to do cf. How does one decompose the regular representation of the group
In my case I am really interested in a representation of dimension 2 (but for an abstract G). So if this helps feel free to assume this.
 A: This may go without saying, but I'd like to first mention that Wedderburn-Artin gives this decomposition as part of the theorem: it says
$$\mathbb{C}[G]\cong M_{n_1}(\mathbb{C})\times\cdots\times M_{n_r}(\mathbb{C})$$
Now each irrep of $G$ (say with a left action) has isotypic component in $\mathbb{C}[G]$ given by one of these factors, $M_{n_i}(\mathbb{C})$ for some $i$.  It follows that to each irrep of $G$ we get a central idempotent $e_i$ in the group ring such that $\mathbb{C}[G]e_i=M_{n_i}(\mathbb{C})$.  In particular, if you can find this central idempotent then you can generate the isotypic component of this irrep inside $\mathbb{C}[G]$.
Now, if that doesn't make you happy, then another thing I can think of is that for any rep $V$ of $G$, each irrep $\pi$ of $G$ has a canonical projector $\sigma_\pi:V\to V$ which projects $V$ onto the $\pi$-isotypic component of $V$.  Applying this construction to $\mathbb{C}[G]$ would allow you to construct the $\pi$-isotypic component in $\mathbb{C}[G]$.
Explicitly, if $\chi_{\pi}$ is the character of $\pi$, then this projector is given by the formula:
$$\sigma_\pi=\frac{n_\pi}{|G|}\sum\limits_{g\in G}\chi_{\pi}(g^{-1})g$$
where $n_\pi=\dim_\mathbb{C}\pi$.  In fact, as an element of the group ring this is just $e_\pi$, the central idempotent associated to $\pi$ (which is the only thing it could be if you think about it).
