Mathematics Lecture Notes

DefinitionComplement

Let $V$ be a representation and $W \subseteq V$ a subrepresentation. A complement to $W$ is a subrepresentation $W' \subseteq V$ such that: $V = W \oplus W'$ as a direct sum of vector spaces and representations.

Not every subrepresentation has a complement in general representation theory.

ExampleWhen Complements Exist

Unitary case: If $V$ has a $G$ -invariant inner product and $W \subseteq V$ is a subrepresentation, then the orthogonal complement $W^\perp = \{v \in V : \langle v, w \rangle = 0 \text{ for all } w \in W\}$ is a complementary subrepresentation.

Finite groups in characteristic 0: By Maschke's theorem, every subrepresentation has a complement.

Counterexample: Consider $\mathbb{Z}$ acting on $\mathbb{C}^2$ via $n \cdot (x,y) = (x, y + nx)$ . The $x$ -axis is an invariant subspace, but it has no complementary invariant subspace (any complement would require a line through $(0,1)$ to be invariant, which is impossible).

TheoremComplement Construction via Averaging

Let $G$ be a finite group, $\mathbb{F}$ a field with $\text{char}(\mathbb{F}) \nmid |G|$ , and $V$ a representation with subrepresentation $W \subseteq V$ . Then $W$ has a complement.

Proof: Choose any vector space complement $W'_0$ (not necessarily invariant). Let $\pi_0: V \to W$ be the projection along $W'_0$ . Define: $\pi = \frac{1}{|G|} \sum_{g \in G} \rho(g) \circ \pi_0 \circ \rho(g^{-1})$

This is a $G$ -equivariant projection onto $W$ . Then $W' = \ker(\pi)$ is a complementary subrepresentation.

DefinitionSemisimple Representation

A representation is called semisimple (or completely reducible) if every subrepresentation has a complement.

Equivalently, $V$ is semisimple if and only if it is a direct sum of irreducible representations.

Remark

The existence of complements is not automatic. It depends on:

The group: Finite groups in characteristic 0 always give complements; infinite groups may not
The field: Characteristic must not divide $|G|$ for finite groups
The representation: Some infinite-dimensional representations never have complements

The failure of complete reducibility in modular representation theory (when $\text{char}(\mathbb{F})$ divides $|G|$ ) leads to a rich and complex theory involving blocks, defect groups, and derived categories.

ExampleDecomposing Representations

For the symmetric group $S_3$ over $\mathbb{C}$ , every representation decomposes into irreducibles:

Trivial: $V_{\text{triv}} = \{(x,x,x) : x \in \mathbb{C}\}$ in the permutation representation
Sign: $V_{\text{sign}} = \{(x,y,z) : x + y + z = 0, \sigma \cdot v = \text{sgn}(\sigma) v\}$
Standard: $V_{\text{std}} = \{(x,y,z) : x + y + z = 0\}$ (two-dimensional)

The permutation representation $\mathbb{C}^3$ decomposes as $\mathbb{C}^3 = V_{\text{triv}} \oplus V_{\text{std}}$ .

Math Notes

Complementary Subrepresentations