# nLab tensor product of representations

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# 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 11:53:35. See the history of this page for a list of all contributions to it.