Complete reducibility over ℂ

Let $G$ be a finite group and $V$ a finite-dimensional complex representation of $G$ . Then $V$ is completely reducible: there are irreducible subrepresentations $V_1,\dots,V_m$ such that

V \;\cong\; V_1\oplus\cdots\oplus V_m.

Equivalently, every subrepresentation $W\subseteq V$ has a $G$ -stable complement $W'\subseteq V$ with

V \;=\; W \oplus W'.

Standard mechanism (unitary averaging)

Choose any Hermitian inner product $\langle\cdot,\cdot\rangle$ on $V$ and average it over $G$ :

\langle v,w\rangle_G \;=\; \frac{1}{|G|}\sum_{g\in G}\langle g\cdot v,\, g\cdot w\rangle.

Then $\langle\cdot,\cdot\rangle_G$ is $G$ -invariant. If $W\subseteq V$ is $G$ -stable, its orthogonal complement $W^\perp$ is also $G$ -stable, giving $V=W\oplus W^\perp$ .

This is a complex-analytic presentation of Maschke's theorem.

Examples

Permutation representation of $S_3$ on $\mathbb C^3$ . Let $S_3$ act by permuting coordinates of $V=\mathbb C^3$ . The line $U=\operatorname{span}\{(1,1,1)\}$ is $S_3$ -stable (it is the trivial representation). The subspace $W=\{(x_1,x_2,x_3)\in\mathbb C^3 : x_1+x_2+x_3=0\}$ is also $S_3$ -stable and $V=U\oplus W$ . Moreover, $W$ is the $2$ -dimensional irreducible (standard) representation.

Any representation of a cyclic group $C_n$ . If $C_n=\langle g\rangle$ and $\rho(g)^n=I$ , then the minimal polynomial of $\rho(g)$ divides $x^n-1$ , which has distinct roots over $\mathbb C$ . Hence $\rho(g)$ is diagonalizable, and $V$ decomposes as a direct sum of eigenspaces. Each eigenspace is a subrepresentation on which $g$ acts by an $n$ th root of unity; choosing a basis in each eigenspace then decomposes it into $1$ -dimensional subrepresentations (characters of $C_n$ ).

The swap representation of $C_2$ on $\mathbb C^2$ . Let $C_2=\{1,s\}$ act on $V=\mathbb C^2$ by $s(x,y)=(y,x)$ . Then $V = \operatorname{span}\{(1,1)\}\;\oplus\;\operatorname{span}\{(1,-1)\}.$ The first summand is the trivial representation; the second is the sign representation (where $s$ acts as $-1$ ).