Direct sum decomposition of $\mathbb CG$ module into simple modules given the character table of $G$ (regular representation)

Question

Suppose I have obtained the character table of a group like $S_3$. After that how can I obtain the simple submodules of $\mathbb CS_3$? It's not clear to me how to go from the irreducible characters of a group to explicitly decomposing its $\mathbb CG$ module.

Answer 1

This is explained in Fulton-Harris, Section 3.4. The point is the following: We know that $\mathbb{C}G\simeq \bigoplus_i\mathrm{End}(W_i)$ , where the $W_i$ are the irreducible representations of $G$. Moreover, the basic properties of characters (and more generally class functions) show that $$ \frac{\dim W_i}{\# G}\sum_g\overline{\chi_{W_i}(g)}e_g\in\mathbb{C}G $$ is an idempotent, and defines a projection onto $W_i^{\oplus\dim W_i}$. To get a projector onto a single summand $W_i$ is much trickier; for symmetric groups, the corresponding idempotents are the Young symmetrizers, but in fact it's not in general known how to construct an irreducible representation just from being given a conjugacy class.