Editing Regular representation (section)

==Module theory point of view==

To put the construction more abstractly, the [[group ring]] ''K''[''G''] is considered as a module over itself. (There is a choice here of left-action or right-action, but that is not of importance except for notation.) If ''G'' is finite and the [[characteristic (algebra)|characteristic]] of K doesn't divide |''G''|, this is a [[semisimple ring]] and we are looking at its left (right) [[ring ideal]]s. This theory has been studied in great depth. It is known in particular that the direct sum decomposition of the regular representation contains a representative of every [[isomorphism class]] of irreducible linear representations of ''G'' over ''K''. You can say that the regular representation is ''comprehensive'' for representation theory, in this case. The modular case, when the characteristic of ''K'' does divide |''G''|, is harder mainly because with ''K''[''G''] not semisimple, a representation can fail to be irreducible without splitting as a direct sum.