Completely reducible linear representation: Difference between revisions
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===Symbol-free 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 linear representations|direct sum]] of [[irreducible linear representation]]s. | A [[linear representation]] of a [[group]] is said to be '''completely reducible''' if it can be expressed as a [[direct sum of linear representations|direct sum]] of [[irreducible linear representation]]s. It may also be called a '''semisimple representation'''. | ||
==Examples== | ==Examples== | ||
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==Non-examples== | ==Non-examples== | ||
<math>(\Z, +)</math> has representations that are not completely reducible - for example <math>\rho</math> such that <math>\rho(N)=\begin{pmatrix} | <math>(\Z, +)</math> has representations that are not completely reducible - for example the representation <math>\rho</math> such that <math>\rho(N)=\begin{pmatrix} | ||
1 & N\\ | 1 & N\\ | ||
0 & 1 | 0 & 1 | ||
\end{pmatrix}</math> can be shown to be a representation which is not completely reducible. | \end{pmatrix}</math> for <math>N \in \mathbb{Z}</math> can be shown to be a representation which is ''not'' completely reducible. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 18:47, 23 October 2023
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
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Non-examples
has representations that are not completely reducible - for example the representation such that for can be shown to be a representation which is not completely reducible.
