If a representation $V$ of a group $G$ is irreducible iff $(\chi_V,\chi_V)=\sum_{i=1}^k a_i^2$=1, where $a_i$ is the multiciplity of each $V_i$ in the isotypic decomposition of $V$, wouldn’t that mean that there’s only one $a_i$ and $a_i$=1?
I’m obviously misunderstanding something. I thought that if $W$ is an irreducible representation of a group $G$, then it would appear as a term in the regular representation of $G$ with multiciplity equal to its dimension $\dim(W)$. But then I don’t see why we wouldn’t have $\dim(W)=1$ for all irreducible representations, which doesn’t seem to be the case… where is my confusion laying?