Irreducible linear representation

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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

A linear representation of a group is said to be irreducible if the vector space being acted upon is a nonzero vector space and there is no proper nonzero invariant subspace for it.

Facts

For finite groups over arbitrary fields

For finite groups, the following are true:

Fore finite groups over a field whose characteristic does not divide the order of the group

  • Maschke's averaging lemma shows that every linear representation is expressible as a direct sum of irreducible linear representations.
  • Orthogonal projection formula gives a concrete method for using the character of a representation to figure out how it decomposes into irreducible representations (note: the formula is simplest in the case of splitting fields)

For finite groups over a splitting field

A splitting field is a field of characteristic not dividing the order of the group whereby the irreducible representations over that field remain irreducible in all bigger fields. For a finite group, \mathbb{C} (the field of complex numbers) and \overline{\mathbb{Q}} are examples of splitting fields. Over a splitting field, we have the following:

In addition, there are a number of other combinatorial and arithmetic controls on the nature and degrees of irreducible representations. For more, see degrees of irreducible representations.

Relation with other properties

Stronger properties

Weaker properties