nLab
tensor product of representations

Contents

Context

Representation theory

Monoidal categories

monoidal categories

With symmetry

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

Idea

The tensor product of linear representations.

Definition

Let GG be a group and let

ρ i:G×V iV i \rho_i \;\colon\; G \times V_i \longrightarrow V_i

be two linear representations of GG on vector spaces V iV_i, for i{1,2}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 kV 2V_1 \otimes_k V_2 equipped with the GG-action induced by the diagonal action

G×V 1×V 2Δ G×idG×G×V 1×V 2G×V 1×G×V 1ρ 1×ρ 2V 1×V 2. 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.