Reducibility Group representation

a subspace w of v invariant under group action called subrepresentation. if v has 2 subrepresentations, namely zero-dimensional subspace , v itself, representation said irreducible; if has proper subrepresentation of nonzero dimension, representation said reducible. representation of dimension 0 considered neither reducible nor irreducible, number 1 considered neither composite nor prime.

under assumption characteristic of field k not divide size of group, representations of finite groups can decomposed direct sum of irreducible subrepresentations (see maschke s theorem). holds in particular representation of finite group on complex numbers, since characteristic of complex numbers zero, never divides size of group.

in example above, first 2 representations given both decomposable 2 1-dimensional subrepresentations (given span{(1,0)} , span{(0,1)}), while third representation irreducible.