Completely reducible linear representation

This article describes a property to be evaluated for a linear representation of a group, i.e. a homomorphism from the group to the general linear group of a vector space over a field

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

Symbol-free definition

A linear representation of a group is said to be completely reducible if it can be expressed as a direct sum of irreducible linear representations. It may also be called a semisimple representation.

Examples

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Non-examples

$(\mathbb {Z} ,+)$ has representations that are not completely reducible - for example the representation $\rho$ such that $\rho (N)={\begin{pmatrix}1&N\\0&1\end{pmatrix}}$ for $N\in \mathbb {Z}$ can be shown to be a representation which is not completely reducible.

Relation with other properties

Stronger properties

• Irreducible linear representation
• Isotypical linear representation