Morphisms and Subrepresentations

Understanding the structure of representations requires studying maps between them that preserve the group action, as well as identifying invariant subspaces.

DefinitionIntertwining Map (Morphism of Representations)

Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be representations of a group $G$ . A linear map $\phi: V_1 \to V_2$ is called an intertwining map or morphism of representations (also called a $G$ -homomorphism) if it commutes with the group action: $\phi(\rho_1(g)(v)) = \rho_2(g)(\phi(v))$ for all $g \in G$ and $v \in V_1$ .

The set of all intertwining maps is denoted $\text{Hom}_G(V_1, V_2)$ and forms a vector space.

ExampleTypes of Morphisms

An intertwining map that is bijective is called an isomorphism of representations
If $\text{Hom}_G(V_1, V_2) = \{0\}$ , the representations are called disjoint
The identity map $\text{id}_V: V \to V$ is always an intertwining map for any representation $V$
For the trivial representation on $\mathbb{F}$ , $\text{Hom}_G(\mathbb{F}, V) = V^G$ (the $G$ -invariant subspace)

DefinitionSubrepresentation

Let $(\rho, V)$ be a representation of $G$ . A subspace $W \subseteq V$ is called a subrepresentation (or $G$ -submodule) if it is invariant under the group action: $\rho(g)(W) \subseteq W \text{ for all } g \in G$

If $W$ is a subrepresentation, the restriction $\rho|_W: G \to GL(W)$ defines a representation on $W$ .

Every representation $V$ has at least two subrepresentations: the trivial subspace $\{0\}$ and $V$ itself. These are called improper subrepresentations.

DefinitionIrreducible and Completely Reducible

A non-zero representation $V$ is called irreducible (or simple) if it has no proper non-zero subrepresentations.

A representation $V$ is called completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations: $V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_k$ where each $V_i$ is irreducible.

Remark

Irreducible representations are the "atoms" of representation theory. A central goal is to classify all irreducible representations of a given group and understand how arbitrary representations decompose into irreducibles. For finite groups over $\mathbb{C}$ , Maschke's theorem guarantees that every representation is completely reducible.

The category-theoretic perspective views representations as objects and intertwining maps as morphisms, creating the category $\text{Rep}(G)$ of $G$ -representations. This framework reveals deep structural properties and connections to other areas of mathematics.