Direct Sums and Tensor Products

Representations can be combined using fundamental constructions from linear algebra, extended naturally to preserve the group action structure.

DefinitionDirect Sum of Representations

Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be representations of $G$ . Their direct sum is the representation $(\rho, V_1 \oplus V_2)$ where: $\rho(g)(v_1, v_2) = (\rho_1(g)(v_1), \rho_2(g)(v_2))$ for all $g \in G$ , $v_1 \in V_1$ , $v_2 \in V_2$ .

This generalizes to finite direct sums $V_1 \oplus \cdots \oplus V_n$ .

ExampleDecomposing Representations

Consider $G = \mathbb{Z}/2\mathbb{Z}$ acting on $\mathbb{R}^2$ by reflection through the origin. This decomposes as $\mathbb{R}^2 = \mathbb{R} e_1 \oplus \mathbb{R} e_2$ where the generator acts as $-1$ on each one-dimensional subspace.

For the regular representation $\mathbb{C}[G]$ of a finite group, it decomposes as: $\mathbb{C}[G] \cong \bigoplus_{V \text{ irreducible}} V^{\oplus \dim V}$ where each irreducible representation appears with multiplicity equal to its dimension.

DefinitionTensor Product of Representations

Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be representations of $G$ . Their tensor product is the representation $(\rho, V_1 \otimes V_2)$ defined on elementary tensors by: $\rho(g)(v_1 \otimes v_2) = \rho_1(g)(v_1) \otimes \rho_2(g)(v_2)$ and extended linearly.

The dimension satisfies $\dim(V_1 \otimes V_2) = \dim(V_1) \cdot \dim(V_2)$ .

The tensor product operation makes $\text{Rep}(G)$ into a monoidal category, with the trivial representation as the unit object.

DefinitionDual Representation

Let $(\rho, V)$ be a representation. The dual representation $(\rho^*, V^*)$ is defined on the dual space $V^* = \text{Hom}_{\mathbb{F}}(V, \mathbb{F})$ by: $(\rho^*(g)(\varphi))(v) = \varphi(\rho(g^{-1})(v))$ for all $g \in G$ , $\varphi \in V^*$ , $v \in V$ .

Remark

The contragredient action (using $g^{-1}$ ) is necessary to ensure $\rho^*$ is a homomorphism: $\rho^*(g_1 g_2) = \rho^*(g_1) \circ \rho^*(g_2)$ . For finite groups, the dual of an irreducible representation is also irreducible, and for unitary representations over $\mathbb{C}$ , the dual is isomorphic to the complex conjugate representation.

These constructions allow us to build complex representations from simpler ones and study the algebraic structure of $\text{Rep}(G)$ . Understanding how irreducible representations decompose under tensor product is a central problem, solved completely for compact Lie groups via the Clebsch-Gordan theory.