# Direct sum of linear representations

## Contents |

This article gives a basic definition in the following area: linear representation theory

View other basic definitions in linear representation theory |View terms related to linear representation theory |View facts related to linear representation theory

## Definition

### Definition in terms of linear representation as a homomorphism

Suppose and are two linear representations of a group over a field . In other words, are vector spaces over and and are homomorphisms. Then, the direct sum is given as follows:

- The vector space is
- Let be the natural map obtained by coordinate-wise action. Then, the homomorphism from to is given by . In other words, the map is:

In matrix terms, it and are finite-dimensional and are given a basis, we can think of as matrix-valued. Then the map is given as:

An analogous definition holds for infinite direct sums. Here, is replaced by a map from the direct product of the s to .

### Definition in terms of linear representation as a group action

Suppose has two linear representations over a field : one on a vector space and the other on the vector space . Then, the direct sum representation on is given by making act coordinate-wise on the direct sum, i.e.:

An analogous coordinate-wise definition holds for infinite direct sums.

### Definition in terms of linear representation as a module over the group ring

Suppose has two linear representations over a field : modules and over the group ring . The direct sum of these representations is the direct sum of and as -modules.

Similarly, we can take an infinite direct sum of linear representations as the infinite direct sum of the corresponding modules.

## Facts

### Effect on character

`Further information: sum of characters equals character of direct sum`

The character of the direct sum of two finite-dimensional linear representations is the sum of their characters. This is because under the map:

The trace of is the sum of the traces of and .

### Effect on determinant

The determinantal representation of the direct sum of two finite-dimensional linear representations is the product of their determinantal representations. That's because, the determinant of is the product of the determinants of and .

### Indecomposable, irreducible, and completely reducible linear representations

`Further information: indecomposable linear representation, irreducible linear representation, completely reducible linear representation`

A linear representation that cannot be expressed as a direct sum of two linear representations on nonzero vector spaces, is termed *indecomposable*.

A stronger notion than indecomposability is the notion of an irreducible linear representation: a linear representation that has no proper nonzero invariant subspace. Maschke's lemma says that in the non-modular case for a finite group (i.e., the case where the characteristic of the field does not divide the order of the group), every indecomposable representation is irreducible.

A representation that can be expressed as a direct sum (possibly infinite) of irreducible linear representations is termed a completely reducible linear representation. It turns out that if every indecomposable linear representation is irreducible, then every representation is completely reducible.