Indecomposable linear representation: Difference between revisions
No edit summary |
No edit summary |
||
| Line 6: | Line 6: | ||
A [[linear representation]] of a [[group]] is said to be '''indecomposable''' if it cannot be expressed as a [[direct sum of linear representations]] with both summands being nonzero (or equivalently, it cannot be expressed as a direct sum of proper nonzero [[subrepresentation]]s). | A [[linear representation]] of a [[group]] is said to be '''indecomposable''' if it cannot be expressed as a [[direct sum of linear representations]] with both summands being nonzero (or equivalently, it cannot be expressed as a direct sum of proper nonzero [[subrepresentation]]s). | ||
Note that in general, the property of being indecomposable is weaker than the property of being irreducible. But [[Maschke's theorem]] tells us that for a [[finite group]] and for a field whose characteristic does not divide the order of the group, every indecomposable representation is indeed irreducible. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 03:06, 27 August 2007
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
Definition
Symbol-free definition
A linear representation of a group is said to be indecomposable if it cannot be expressed as a direct sum of linear representations with both summands being nonzero (or equivalently, it cannot be expressed as a direct sum of proper nonzero subrepresentations).
Note that in general, the property of being indecomposable is weaker than the property of being irreducible. But Maschke's theorem tells us that for a finite group and for a field whose characteristic does not divide the order of the group, every indecomposable representation is indeed irreducible.