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Irreducible linear representation

478 bytes added, 15:16, 1 August 2011
* 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]].
===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===
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