Given an finite group $G$ and its group algebra $\mathbb{C}G$ and the order relation on the idempotents of $\mathbb{C}G$ such that $e\leq f$ iff $ef=fe=e$. Is the following true:
The irreducible characters are in bijection with the minimal idempotents of the center of the algebra $\mathbb{C}G$ .
The center of the algebra is the class function on $G$.
Thanks in advance.
Yes and sort of. Given an irreducible character $\chi$, the corresponding minimal central idempotent, which projects a representation of $G$ to its $\chi$-isotypic component, is
$$\frac{1}{|G|} \sum_{g \in G} \overline{\chi}(g) g.$$
The center of $\mathbb{C}[G]$ has a basis given by sums over conjugacy classes, and the corresponding algebra is in some sense dual to the algebra of class functions. Its spectrum is naturally identified with the irreducible representations of $G$, while the spectrum of the algebra of class functions is naturally identified with the conjugacy classes of $G$.
