Complete Reducibility

The question of when representations decompose into irreducibles is fundamental. Complete reducibility vastly simplifies the classification problem by reducing it to understanding irreducible representations.

TheoremEquivalent Conditions for Complete Reducibility

For a finite-dimensional representation $V$ of $G$ , the following are equivalent:

$V$ is completely reducible (isomorphic to a direct sum of irreducibles)
Every subrepresentation $W \subseteq V$ has a complementary subrepresentation: $V = W \oplus W'$ for some $W' \subseteq V$
$V$ is a sum (not necessarily direct) of irreducible subrepresentations
There exists no infinite chain of proper subrepresentations (ascending or descending chain condition)

The implication (2) $\Rightarrow$ (1) proceeds by induction: if $V$ is not irreducible, choose a proper subrepresentation $W$ , find a complement $W'$ , and decompose each piece recursively.

ExampleComplete Reducibility in Practice

Unitary representations: Every finite-dimensional unitary representation (where $\rho(g)$ are unitary operators) is completely reducible. Given a subrepresentation $W$ , the orthogonal complement $W^\perp$ is automatically a complementary subrepresentation.

Finite groups over $\mathbb{C}$ : By Maschke's theorem, every representation of a finite group in characteristic 0 is completely reducible. This fails in positive characteristic dividing $|G|$ .

Counterexample: The two-dimensional representation of $\mathbb{Z}$ generated by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has the $x$ -axis as an invariant line, but no complementary invariant line—it is not completely reducible.

DefinitionIndecomposable Representation

A non-zero representation $V$ is called indecomposable if it cannot be written as a non-trivial direct sum: if $V = V_1 \oplus V_2$ , then either $V_1 = 0$ or $V_2 = 0$ .

Every irreducible representation is indecomposable, but the converse fails without complete reducibility.

TheoremKrull-Schmidt Theorem

Every finite-dimensional representation decomposes uniquely (up to isomorphism and reordering) into a direct sum of indecomposable representations: $V \cong V_1 \oplus \cdots \oplus V_k$

When complete reducibility holds, "indecomposable" coincides with "irreducible," so this becomes the unique decomposition into irreducibles (with multiplicities).

Remark

Complete reducibility dramatically simplifies representation theory: instead of studying all representations, we need only classify irreducibles and understand their multiplicities. The representation ring (or Grothendieck group) is freely generated by irreducibles, tensor products decompose via explicit formulas (Clebsch-Gordan), and character theory provides a complete set of invariants.

When complete reducibility fails (e.g., modular representations), the theory becomes significantly more complex, involving derived categories, homological methods, and the study of indecomposable representations which are not irreducible.