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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and in particular the theory of
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s, the regular representation of a group ''G'' is the
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
afforded by the
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of ''G'' on itself by
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation.


Finite groups

For a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
''G'', the left regular representation λ (over a field ''K'') is a linear representation on the ''K''-vector space ''V'' freely generated by the elements of ''G'', i. e. they can be identified with a basis of ''V''. Given ''g'' ∈ ''G'', λ''g'' is the linear map determined by its action on the basis by left translation by ''g'', i.e. :\lambda_:h\mapsto gh,\texth\in G. For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g'' ∈ ''G'', ρ''g'' is the linear map on ''V'' determined by its action on the basis by right translation by ''g''−1, i.e. :\rho_:h\mapsto hg^,\texth\in G.\ Alternatively, these representations can be defined on the ''K''-vector space ''W'' of all functions . It is in this form that the regular representation is generalized to
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s such as
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s. The specific definition in terms of ''W'' is as follows. Given a function and an element ''g'' ∈ ''G'', :(\lambda_f)(x)=f(\lambda_^(x))=f(^x) and :(\rho_f)(x)=f(\rho_^(x))=f(xg).


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 In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...
it is characterised as having a single
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
and stabilizer the identity subgroup 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 In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
field, the regular representation decomposes as a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
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 In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es of ''G''. The above fact can be explained by
character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
. 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''''i''''V''''i'' where ''V''''i'''s are irreducible representations of ''G'' and ''a''''i'''s are the corresponding multiplicities. By
character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
, the multiplicity ''a''''i'' can be computed as a_i= \langle \chi,\chi_i \rangle =\frac\sum \overline\chi_i(g)=\frac\chi(1)\chi_i(1)=\operatorname V_i, which means the multiplicity of each irreducible representation is its dimension. The article on
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
s articulates the regular representation for
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
s, as well as showing how the regular representation can be taken to be a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
.


Module theory point of view

To put the construction more abstractly, the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
''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 In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
and we are looking at its left (right)
ring ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers p ...
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.


Structure for finite cyclic groups

For a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
''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 linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplit ...
'', in which each row is a shift to the right of the one above (in
cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...
, 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 can diagonalise the representation of ''C'' by writing down ''n'' linearly independent simultaneous
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
''K'' (such as the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s) 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 mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
''. 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s 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 representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
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 Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, the factorisation must contain non-linear factors corresponding to
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
s of ''G'' of degree > 1.


Topological group case

For a topological group ''G'', 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 In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
for the
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
case. If ''G'' is a Lie group but not compact nor
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, this is a difficult matter of
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
. The
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
abelian case is part of the
Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
theory.


Normal bases in Galois theory

In
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
it is shown that for a field ''L'', and a finite group ''G'' of
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
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 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
it is of interest to study ''normal integral bases'', where we try to replace ''L'' and ''K'' by the rings of
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s they contain. One can see already in the case of the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
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 in
Galois module In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
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 In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
''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 a
Frobenius algebra In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality th ...
. 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 gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathe ...
in 1 + 1 dimensions by a particular instance of the
cobordism hypothesis In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the deta ...
.


See also

*
Fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
*
Permutation representation In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...
* Quasiregular representation


References

*{{Fulton-Harris Representation theory of groups