Complete reducibility over ℂ

Let $G$ be a finite group and let $V$ be a finite-dimensional complex representation of $G$.

Theorem (complete reducibility over $\mathbb C$)

$V$ is completely reducible : there exist irreducible subrepresentations $V_1,\dots,V_m$ such that

Equivalently: for every subrepresentation $W\subseteq V$, there exists a $G$-stable complement $W'\subseteq V$ with

Standard mechanism (unitary averaging)

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

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

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

Examples

1. 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.

2. 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 $1$-dimensional subrepresentation on which $g$ acts by an $n$th root of unity (a character of $C_n$).

3. 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 $-1$).