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Irreducible linear representation

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==Definition==
===Symbol-free 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 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.==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 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]].
 
===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. 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]].
==Relation with other properties==
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