# nLab tensor product of representations

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Monoidal categories

monoidal categories

# Contents

## 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

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

Last revised on January 22, 2019 at 06:53:35. See the history of this page for a list of all contributions to it.