Projection formula

General projection formula Let $U=V_1^{\oplus a_1}\oplus \cdots V_k^{\oplus a_k}$ with $V_i$ distinct irreducibles. Then ${\phi }_i=\dim V_i\frac{1}{|G|}{\sum }_{g\in G}\overline{{\chi }_V(g)}(g)\in \mathrm{End}(U)$ is the projection $U\to V_i^{\oplus a_i}$