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 (or sometimes simple) if the vector space being acted upon is a nonzero vector space and there is no proper nonzero invariant subspace for it. That is, it has no proper subrepresentations.
Facts
For finite groups over arbitrary fields
For finite groups, the following are true:
- Over any field, there are only finitely many irreducible representations, and there is a bound on the degree (the dimension of the vector space acted upon): degree of irreducible representation of nontrivial finite group is strictly less than order of group.
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, (the field of complex numbers) and are examples of splitting fields. Over a splitting field, we have the following:
- Number of irreducible representations equals number of conjugacy classes
- Sum of squares of degrees of irreducible representations equals order of group, regular representation over splitting field has multiplicity of each irreducible representation equal to its degree, Group ring over splitting field is direct sum of matrix rings for each irreducible representation
- Character orthogonality theorem, column orthogonality theorem, splitting implies characters separate conjugacy classes, splitting implies characters form a basis for space of class functions
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.