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.
Contents 
Significance of the regular representation of a group
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.
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 leftaction or rightaction, 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.
Structure for finite cyclic groups
For G a cyclic group C generated by g of order n, the matrix form of an element of K[C] acting on K[C] by multiplication takes a distinctive form known as a circulant matrix, in which each row is a shift to the right of the one above (in cyclic order, i.e. with the rightmost element appearing on the left), when referred to the natural basis
 1, g, g^{2}, ..., g^{n1}.
When the field K contains a primitive nth root of unity, one can diagonalise the representation of C by writing down n linearly independent simultaneous eigenvectors for all the n×n circulants. In fact if ζ is any nth root of unity, the element
 1+ζg+ζ^{2}g^{2}+ ... +ζ^{n1}g^{n1}
is an eigenvector for the action of g by multiplication, with eigenvalue
 ζ^{1}
and so also an eigenvector of all powers of g, and their linear combinations.
This is the explicit form in this case of the abstract result that over an algebraically closed field K (such as the complex numbers) the regular representation of G is completely reducible, provided that the characteristic of K (if it is a prime number p) doesn't divide the order of G. That is called Maschke's theorem. In this case the condition on the characteristic is implied by the existence of a primitive nth root of unity, which cannot happen in the case of prime characteristic p dividing n.
Circulant determinants were first encountered in nineteenth century mathematics, and the consequence of their diagonalisation drawn. Namely, the determinant of a circulant is the product of the n eigenvalues for the n eigenvectors described above. The basic work of Frobenius on group representations started with the motivation of finding analogous factorisations of the group determinants for any finite G; that is, the determinants of arbitrary matrices representing elements of K[G] acting by multiplication on the basis elements given by g in G. Unless G is abelian, the factorisation must contain nonlinear factors corresponding to irreducible representations of G of degree > 1.
Topological group case
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 PeterWeyl 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.
Normal bases in Galois theory
In Galois theory it is shown that for a field L, and a finite group of automorphisms of L, the fixed field K of G has [L:K] = G. In fact we can say more: L as K[G]module is the regular representation. This is the content of the normal basis theorem, a normal basis being an element x of L such that the g(x) for g in G are a vector space basis for L over K. Such x exist, and each one gives a K[G]isomorphism from L to K[G]. From the point of view of algebraic number theory it is of interest to study normal integral bases, where we try to replace L and K by the rings of algebraic integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: a+bi and abi can never form a Zmodule basis of Z[i] because 1 cannot be an integer combination. The reasons are studied in depth in Galois module theory.
More general algebras
The regular representation of a group ring is such that the lefthand and righthand regular representations give isomorphic modules (and we often need not distinguish the cases). Given an algebra over a field A, it doesn't immediately make sense to ask about the relation between A as leftmodule over itself, and as rightmodule. In the group case, the mapping on basis elements g of K[G] defined by taking the inverse element gives an isomorphism of K[G] to its opposite ring. For A general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been shown to be related to topological quantum field theory in 1+1 dimensions.