Editing Regular representation (section)

==Significance of the regular representation of a group==

Every group ''G'' acts on itself by translations. If we consider this action as a [[permutation representation]] it is characterised as having a single [[orbit (group theory)|orbit]] and [[Group action (mathematics)|stabilizer]] the identity subgroup {''e''} of ''G''. The regular representation of ''G'', for a given field ''K'', is the linear representation made by taking this permutation representation as a set of [[basis vector]]s of a [[vector space]] over ''K''. The significance is that while the permutation representation doesn't decompose – it is [[Group action (mathematics)|transitive]] – the regular representation in general breaks up into smaller representations. For example, if ''G'' is a finite group and ''K'' is the [[complex number]] field, the regular representation decomposes as a [[direct sum of representations|direct sum]] of [[irreducible representation]]s, with each irreducible representation appearing in the decomposition with multiplicity its dimension. The number of these irreducibles is equal to the number of [[conjugacy class]]es of ''G''.

The above fact can be explained by [[character theory]]. Recall that the character of the regular representation χ''(g)'' is the number of fixed points of ''g'' acting on the regular representation ''V''. It means the number of fixed points χ''(g)'' is zero when ''g'' is not ''id'' and |''G''| otherwise. Let ''V'' have the decomposition ⊕''a''<sub>''i''</sub>''V''<sub>''i''</sub> where ''V''<sub>''i''</sub>'s are irreducible representations of ''G'' and ''a''<sub>''i''</sub>'s are the corresponding multiplicities. By [[character theory]], the multiplicity ''a''<sub>''i''</sub> can be computed as 

<math>a_i= \langle \chi,\chi_i \rangle =\frac{1}{|G|}\sum \overline{\chi(g)}\chi_i(g)=\frac{1}{|G|}\chi(1)\chi_i(1)=\operatorname{dim} V_i,</math>
which means the multiplicity of each irreducible representation is its dimension.

The article on [[group ring]]s articulates the regular representation for [[finite group]]s, as well as showing how the regular representation can be taken to be a [[module (mathematics)|module]].