- Regular representation
In
mathematics , and in particular the theory ofgroup representation s, the regular representation of a group "G" is thelinear representation afforded by thegroup action of "G" on itself.ignificance 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 ofbasis vector s of avector 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 thecomplex number field, the regular representation is adirect sum ofirreducible representation s, in number at least the number ofconjugacy class es of "G".The article on
group algebra s articulates the regular representation forfinite group s, as well as showing how the regular representation can be taken to be a module.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 of K doesn't divide |"G"|, this is asemisimple 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.tructure for finite cyclic groups
For 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 (incyclic order , i.e. with the right-most element appearing on the left), when referred to the natural basis:1, "g", "g"2, ..., "g""n"−1.
When the field "K" contains a
primitive n-th root of unity , one candiagonalise the representation of "C" by writing down "n" linearly independent simultaneouseigenvector s for all the "n"×"n" circulants. In fact if ζ is any "n"-th root of unity, the element:1 + ζ"g" + ζ2"g"2 + ... + ζ"n"−1"g""n"−1
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 thecomplex number s) the regular representation of "G" iscompletely 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" "n"-th root of unity, which cannot happen in the case of prime characteristic "p" dividing "n".Circulant
determinant s were first encountered innineteenth 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 ongroup representation s 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 non-linear factors corresponding toirreducible representation s 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. SeePeter-Weyl theorem for thecompact case. If "G" is aLie group but not compact nor abelian, this is a difficult matter ofharmonic analysis . Thelocally compact abelian case is part of thePontryagin duality theory.Normal bases in Galois theory
In
Galois theory it is shown that for a field "L", and a finite group "G" ofautomorphism s of "L", the fixed field "K" of "G" has ["L":"K"] = |"G"|. In fact we can say more: "L" viewed as a "K" ["G"] -module is the regular representation. This is the content of thenormal basis theorem , a normal basis being an element "x" of "L" such that the "g"("x") for "g" in "G" are avector 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 ofalgebraic number theory it is of interest to study "normal integral bases", where we try to replace "L" and "K" by the rings ofalgebraic integer s they contain. One can see already in the case of theGaussian integer s that such bases may not exist: "a" + "bi" and "a" − "bi" can never form a Z-module basis of Z ["i"] because 1 cannot be an integer combination. The reasons are studied in depth inGalois module theory.More general algebras
The regular representation of a group ring is such that the left-hand and right-hand 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 left-module over itself, and as right-module. 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 aFrobenius algebra . As the name implies, these were introduced by Frobenius in the nineteenth century. They have been shown to be related totopological quantum field theory in 1 + 1 dimensions.
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