nLab tensor product of representations

Idea

The tensor product of linear representations.

Definition

Let $G$ be a group and let

$$\rho_i \;\colon\; G \times V_i \longrightarrow V_i$$

be two linear representations of $G$ on vector spaces $V_i$, for $i \in \{1,2\}$. Then the tensor product of representations of these is the linear representation whose underlying vector space is the tensor product of vector spaces $V_1 \otimes_k V_2$ equipped with the $G$-action induced by the diagonal action

$$G \times V_1 \times V_2 \overset{\Delta_G \times id}{\longrightarrow} G \times G \times V_1 \times V_2 \simeq G \times V_1 \times G \times V_1 \overset{\rho_1 \times \rho_2}{\longrightarrow} V_1 \times V_2 \,.$$

Related concepts

• category of representations

• Clebsch-Gordan coefficient

• Wigner 3j symbol, Wigner 6j symbol, Wigner 9j symbol

• McKay quiver