Completely reducible representation

Definition

Let $(V,\rho)$ be a finite-dimensional group representation of $G$ over a field $k$. The representation $V$ is completely reducible if there exist irreducible subrepresentations $V_1,\dots,V_r$ such that

as $G$-representations (i.e. as $k[G]$-modules). Here $\oplus$ is the direct sum in the module/representation sense.

Equivalently, $V$ is completely reducible iff every subrepresentation $W\subseteq V$ has a $G$-stable complement: there exists a subrepresentation $U\subseteq V$ such that

In module language, this is exactly: $V$ is a semisimple module over the group algebra $k[G]$.

Maschke’s criterion

If $G$ is finite and $\mathrm{char}(k)\nmid |G|$, then every finite-dimensional $k$-representation of $G$ is completely reducible: this is Maschke’s theorem . In particular, over $k=\mathbb{C}$, all finite-group representations are completely reducible (see complete reducibility over \u211d/\u2102 ).

Examples

1. A completely reducible permutation representation of $S_3$ over $\mathbb{C}$.
   Let $V=\mathbb{C}^3$ with $S_3$ permuting coordinates. Then the subspaces

   $$W_{\mathrm{triv}}=\mathrm{span}\{(1,1,1)\},\qquad W_{\mathrm{std}}=\{(x_1,x_2,x_3):x_1+x_2+x_3=0\}$$

   are $S_3$-stable and

   $$\mathbb{C}^3 = W_{\mathrm{triv}} \oplus W_{\mathrm{std}}.$$

   Here $W_{\mathrm{triv}}$ is trivial and $W_{\mathrm{std}}$ is irreducible .

2. Regular representations over $\mathbb{C}$ split into irreducibles with multiplicities.
   For a finite group $G$, the regular representation $\mathbb{C}[G]$ is completely reducible and decomposes as

   $$\mathbb{C}[G]\ \cong\ \bigoplus_{i} (\dim V_i)\, V_i,$$

   where $\{V_i\}$ runs over the irreducible $\mathbb{C}$-representations of $G$. Taking dimensions yields

   $$|G|=\sum_i (\dim V_i)^2,$$

   i.e. the sum of squares of degrees identity.

3. A non-example in characteristic $p\mid |G|$: no invariant complement.
   Let $G=C_p=\langle g\rangle$ and $k=\mathbb{F}_p$. On $V=k^2$, define

   $$\rho(g)=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

   The line $W=\mathrm{span}\{e_1\}$ is $G$-stable, so it is a subrepresentation. If $V$ were completely reducible, there would be a $G$-stable complementary line $U$ with $V=W\oplus U$. But $\rho(g)$ has only one eigenline in characteristic $p$, namely $W$, so no such $U$ exists. Hence $V$ is not completely reducible.

See also: Maschke corollary , character of a direct sum .