# Tensor product of linear representations

## Contents

## Definition

Suppose is a group and and are linear representations of over a field . The **tensor product** of the representations, denoted is a linear representation on the tensor product of the vector spaces defined in the following equivalent ways.

### Definition using tensor product of linear maps

as follows: for , . Here, is the image of the pair in the natural homomorphism .

Conceptually, the mapping:

is described as follows: we know that up to isomorphism determine up to isomorphism. This means that any choice of automorphism of along with automorphism of induces an automorphism of . The mapping describes how.

### Definition using outer tensor product

The tensor product can be defined as the composite of the outer tensor product of linear representations of and (which gives a linear representation of ) with the diagonal inclusion map , a homomorphism from to .

### Definition in terms of symmetric bimonoidal category

The tensor product of linear representations over a field can be defined as the tensor product of representations over a symmetric bimonoidal category where the category is the category of -vector spaces, the additive operation is direct sum of vector spaces, and the multiplicative operation is tensor product of vector spaces.

### Explicit definition in terms of block matrices

This definition works for finite-dimensional linear representations, though it also has infinite-dimensional analogues if we use infinitary matrices.

We use the same notation as in the previous definition, but assume further that and . Then can be identified with where the first coordinates represent one copy of , the next copies represent the next copy of , and so on. The explicit definition is now given as follows: for , first write the matrix for . Then, replace each cell of the matrix by a matrix that equals the cell value times . Overall, we get a matrix.

## Definition over a commutative unital ring

Suppose is a group and and are linear representations of over a commutative unital ring . The **tensor product** of the representations, denoted is a linear representation on the tensor product of the modules defined as follows: for , . Here, is the image of the pair in the natural homomorphism .

Conceptually, the mapping:

is described as follows: we know that up to isomorphism determine up to isomorphism. This means that any choice of automorphism of along with automorphism of induces an automorphism of . The mapping describes how.

Note that the explicit matrix description of tensor product is available only if the modules and are free modules of finite rank over .

## Related notions

- Generalization: tensor product of representations over a symmetric bimonoidal category
- Tensor product of permutation representations

## Facts

- Degree of tensor product of linear representations is product of degrees
- Character of tensor product of linear representations is product of characters
- Tensor product of irreducible representation and one-dimensional representation is irreducible
- Tensor product of irreducible representations need not be irreducible