= Character projection operator
{title2=$P_\lambda=\frac{d_\lambda}{|G|}\sum_{g\in G}\chi_\lambda(g)^*U(g)$}
For a finite <group> $G$ represented unitarily by $U(g)$, the central projector onto the isotypic subspace of an <irreducible representation> $\lambda$ is $P_\lambda=(d_\lambda/|G|)\sum_g\chi_\lambda(g)^*U(g)$, where $d_\lambda$ is its <dimension>. <Character orthogonality> implies that the operator is identity on copies of $\lambda$ and zero on other irreducibles. For a one-dimensional irreducible, applying it to an <atomic orbital> gives either zero or a <symmetry-adapted linear combination> of that <symmetry>, to be normalized afterward. A multidimensional central projector selects an entire isotypic component rather than one uniquely chosen component function.
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