Isotypic decomposition

A finite-length semisimple module has a canonical decompositionin which $M_i$ is the sum of all simple submodules isomorphic to a fixed simple module $S_i$. Each $M_i$ is isomorphic to a finite direct sum $S_i^{m_i}$.

The decomposition is intrinsic to the semisimple module: the isomorphism types of its simple submodules determine the summands.

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 $N◃G$, conjugation permutes them according to ${\theta }^g(n)=\theta (g^{-1}ng)$.