Difference between revisions of "Irreducible linear representation"

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==Definition==
 
==Definition==
  
===Symbol-free definition===
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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 a linear representation|invariant subspace]] for it.
  
A [[linear representation]] of a [[group]] is said to be '''irreducible''' if there is no proper nonzero [[invariant subspace for a linear representation|invariant subspace]] for it.
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==Facts==
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===For finite groups over arbitrary fields===
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For finite groups, the following are true:
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* Over ''any'' field, there are only finitely many irreducible representations, and there is a bound on the [[degree of a linear representation|degree]] (the dimension of the vector space acted upon): [[degree of irreducible representation of nontrivial finite group is strictly less than order of group]].
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===Fore finite groups over a field whose characteristic does not divide the order of the group===
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* [[Maschke's averaging lemma]] shows that every linear representation is expressible as a direct sum of irreducible linear representations.
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* [[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)
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===For finite groups over a splitting field===
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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]], <math>\mathbb{C}</math> (the field of complex numbers) and <math>\overline{\mathbb{Q}}</math> are examples of splitting fields. Over a splitting field, we have the following:
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* [[Number of irreducible representations equals number of conjugacy classes]]
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* [[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]]
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* [[Character orthogonality theorem]], [[column orthogonality theorem]], [[splitting implies characters separate conjugacy classes]], [[splitting implies characters form a basis for space of class functions]]
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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==
 
==Relation with other properties==

Latest revision as of 15:16, 1 August 2011

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