# Irreducible 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

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.