Regular representation

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself. To say that G acts on itself by multiplication is tautological. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {e} of G. The regular representation of G, for a given field K, is the linear representation made by taking the permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose - it is 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 is a direct sum of irreducible representations, in number at least the number of conjugacy classes of G.

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 of K doesn't divide |G|, this is a semisimple ring and we are looking at its left (right) ring ideals. 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 representation 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.

For G a topological group, the regular representation in the above sense should be replaced by a suitable space of functions on G, with G acting by translation. See Peter-Weyl theorem for the compact case. If G is a Lie group but not compact nor abelian, this is a difficult matter of harmonic analysis. The locally compact abelian case is part of the Pontryagin duality theory.