= Isotypic component
{title2=$V_\theta=e_\theta V$}
In a complex representation of a <finite group> $N$, the $\theta$-isotypic component is the sum of all irreducible submodules with character $\theta$. The <character idempotent> $e_\theta=\theta(1)|N|^{-1}\sum_{n\in N}\theta(n^{-1})n$ projects onto it. Every $N$-submodule is preserved by these projections and decomposes into its intersections with the isotypic components. If $N\triangleleft G$, conjugation permutes them according to $\theta^g(n)=\theta(g^{-1}ng)$.
Back to article page
