Indecomposable 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
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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.

Facts

Number of indecomposable linear representations

A finite group has a finite number of nonisomorphic indecomposable linear representations over a field of characteristic $p$, if and only if its Sylow $p$-subgroup is characteristic. Of course, in the case that $p$ does not divide the order of the group, the indecomposable linear representations are the same as the irreducible linear representations (by Maschke's lemma) and thus there are no more than the number of irreducible characters. In general, however, there could be a very large number of indecomposable linear representations.