Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation isomorphic to $\pi$ (there are $(\dim \pi)^2$ linearly independent such elements.) It should be some standard `canonical' way to do this, but I shamefully do not see it.


Particular case of $S\_4$ and 2-dimensional representation is of interest, but general procedure is appreciated as well. 



how to find explicitly given component in a regular representation
The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$ (see details below).


This follows from the explicit Wedderburn decomposition of $\mathbb C[G]$. If we write
$$
\mathbb C[G] = \bigoplus\_{\pi \in \hat G} M\_{d\_\pi}(\mathbb C),
$$
then the identity element of the $\pi$th summand is 
$$
\epsilon\_\pi = \sum\_{g\in G} \frac{d\_\pi}{|G|}\chi\_\pi(g^{-1})g,
$$
where $\chi\_\pi$ is the character of $\pi$. See for example Theorem 1.7.9 in [my book](http://www.imsc.res.in/~amri/rtcv/), the relevant part of which is also available online [here](http://www.cimpa-icpam.org/IMG/pdf/RepresentationTh.pdf) (see Theorem 7.9).


