I know that every irreducible representations of $S_n$ can be found in $\mathbb{C}S_n$. I wonder how can I prove that irreducible representations of a finite groups $G$ can be found in $\mathbb{C}G$. I thought to use the embedding of $G$ in $S_n$, but then I don't know how to go on. Could any of you help me, please?
Maybe I got it: I know that the character of the regular representation can be viewed as the sum of the characters of the irreducible representations with coefficients their dimension, so they are in the decomposition of the regular representation. Am I right?
Best Answer
To prove this statement using characters is kind of working backwards.
You can prove this statement without characters as follows: first, show that any simple $A=\mathbb{C}[G]$-module $M$ is a quotient of $A$. To do that, fix $0\neq m\in M$ and define $\mathbb{C}[G]\rightarrow M, \; \sum\alpha_gg \mapsto\sum\alpha_gg(m)$. Show that this is onto ($M$ is simple) and use the isomorphism theorem. Now, use Maschke's theorem to show that if $M$ is a quotient of $A$, then it is also a direct summand of $A$.
For details of all this, see e.g. Proposition 2.14 of my notes on representation theory.