Editing Group representation (section)

== Reducibility ==

{{main|Irreducible representation}}

A subspace ''W'' of ''V'' that is invariant under the [[Group action (mathematics)|group action]] is called a ''[[subrepresentation]]''. If ''V'' has exactly two subrepresentations, namely the zero-dimensional subspace and ''V'' itself, then the representation is said to be '''irreducible'''; if it has a proper subrepresentation of nonzero dimension, the representation is said to be '''reducible'''. The representation of dimension zero is considered to be neither reducible nor irreducible, <ref>{{Cite web|date=2019-09-04|title=1.4: Representations|url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Advanced_Inorganic_Chemistry_(Wikibook)/01%3A_Chapters/1.04%3A_Representations|access-date=2021-06-23|website=Chemistry LibreTexts|language=en}}</ref> just as the number 1 is considered to be neither [[Composite number|composite]] nor [[Prime number|prime]].

Under the assumption that the [[characteristic (algebra)|characteristic]] of the field ''K'' does not divide the size of the group, representations of [[finite group]]s can be decomposed into a [[direct sum of groups|direct sum]] of irreducible subrepresentations (see [[Maschke's theorem]]). This holds in particular for any representation of a finite group over the [[complex numbers]], since the characteristic of the complex numbers is zero, which never divides the size of a group.

In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation (τ) is irreducible.