I have some trouble understanding the decomposition of representations into irreducible ones.
For example, take $G = S_3$, the symmetric group.
Then $G$ has three irreducible representations, namely, the trivial one, the sign one, and the two-dimensional represention given by the planar symmetries of a triangle.
Let $V$ be the $\mathbb C G$-module given by the two-dimensional irreducible representation, $T$ be the trivial module, and $S$ be the module given by the sign representation.
It is easy to see from the characters that $V\otimes V \cong V\oplus T\oplus S$ as $\mathbb C G$-modules. However, what explicitly is this $G$-invariant isomorphism?