Representations and Modules

Representation theory provides a powerful bridge between abstract algebraic structures and linear algebra, allowing us to study groups, algebras, and other mathematical objects through their actions on vector spaces.

DefinitionGroup Representation

Let $G$ be a group and $V$ a vector space over a field $\mathbb{F}$ . A representation of $G$ on $V$ is a group homomorphism $\rho: G \to GL(V)$ where $GL(V)$ denotes the group of invertible linear transformations of $V$ . The dimension of $V$ is called the degree of the representation.

Equivalently, a representation assigns to each $g \in G$ a linear map $\rho(g): V \to V$ such that:

$\rho(e) = \text{id}_V$ (identity element maps to identity transformation)
$\rho(g_1 g_2) = \rho(g_1) \circ \rho(g_2)$ for all $g_1, g_2 \in G$

ExampleClassical Examples

Trivial representation: $\rho(g) = \text{id}_V$ for all $g \in G$
Regular representation: For finite $G$ , the vector space $\mathbb{C}[G]$ with basis elements $\{e_g : g \in G\}$ and action $\rho(h)(e_g) = e_{hg}$
Permutation representation: For $G = S_n$ acting on $\mathbb{R}^n$ , permuting coordinates
Matrix groups: $GL_n(\mathbb{C})$ acts on $\mathbb{C}^n$ by matrix multiplication (the defining representation)

DefinitionG-Module

A $G$ -module is a vector space $V$ together with a linear action of $G$ on $V$ , written as $g \cdot v$ for $g \in G$ and $v \in V$ , satisfying:

$e \cdot v = v$ for all $v \in V$
$g_1 \cdot (g_2 \cdot v) = (g_1 g_2) \cdot v$ for all $g_1, g_2 \in G$ and $v \in V$
$g \cdot (\alpha v_1 + \beta v_2) = \alpha(g \cdot v_1) + \beta(g \cdot v_2)$ for all $g \in G$ , $v_1, v_2 \in V$ , $\alpha, \beta \in \mathbb{F}$

The concepts of representation and $G$ -module are equivalent: given a representation $\rho: G \to GL(V)$ , we define a module structure by $g \cdot v = \rho(g)(v)$ . Conversely, a $G$ -module structure determines a representation $\rho(g)(v) = g \cdot v$ .

Remark

The module perspective is often more natural when working with algebras and studying structural properties, while the homomorphism perspective emphasizes the functorial nature of representations and facilitates the study of morphisms between representations.

Representations allow us to translate abstract group-theoretic questions into concrete problems in linear algebra, where powerful computational and theoretical tools are available. The fundamental problems include classifying all representations, understanding how they decompose into simpler pieces, and computing invariants that distinguish non-isomorphic representations.