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Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.

Please note, that except for a few marked exceptions only finite groups will be considered in this article. We will also restrain to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over

Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms.

Contents

DefinitionEdit

Linear representationsEdit

Let   be a  –vector space and   a finite group. A linear representation of a finite group   is a group homomorphism   That means, a linear representation is a map   which satisfies   for all   The vector space   is called representation space of   Often the term representation of   is also used for the representation space  

The representation of a group in a module instead of a vector space is also called a linear representation.

We write   for the representation   of   Sometimes we only use   if it is clear to which representation the space   belongs.

In this article we will restrain ourselves to the study of finite-dimensional representation spaces, except for the last chapter. As in most cases only a finite number of vectors in   is of interest, it is sufficient to study the subrepresentation generated by these vectors. The representation space of this subrepresentation is then finite-dimensional.

The degree of a representation is the dimension of its representation space   The notation   is sometimes used to denote the degree of a representation  

ExamplesEdit

The trivial representation is given by   for all  

A representation of degree   of a group   is a homomorphism into the multiplicative group   As every element of   is of finite order, the values of   are roots of unity. For example let   be a nontrivial linear representation. Since   is a group homomorphism, it has to satisfy   Because   generates   is determined by its value on   And as   is nontrivial,   By this we achieve the result, that the image of   under   has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. This means, that   has to be one of the following three maps:

 

Let   and let   be the group homomorphism defined by:

 

In this case   is a linear representation of   of degree  

Permutation representationEdit

Let   be a finite set. Let   be a group operating on   The group   is the group of all permutations on   with the composition as operation.

A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations, where groups act on vector spaces instead of on arbitrary finite sets, we have to proceed in a different way. In order to construct the permutation representation, we need a vector space   with   A basis of   can be indexed by the elements of   The permutation representation is the group homomorphism   given by   for all   All linear maps   are uniquely defined by this property.

Example. Let   and   Then   operates on   via   The associated linear representation is   with   for  

Left- and right-regular representationEdit

Let   be a group and   be a vector space of dimension   with a basis   indexed by the elements of   The left-regular representation is a special case of the permutation representation by choosing   This means   for all   Thus, the family   of images of   are a basis of   The degree of the left-regular representation is equal to the order of the group.

The right-regular representation is defined on the same vector space with a similar homomorphism:   In the same way as before   is a basis of   Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order of  

Both representations are isomorphic via   For this reason they are not always set apart, and often referred to as the regular representation.

A closer look provides the following result: A given linear representation   is isomorphic to the left-regular representation, if and only if there exists a   such that   is a basis of  

Example. Let   and   with the basis   Then the left-regular representation   is defined by   for   The right-regular representation is defined analogously by   for  

Representations, modules and the convolution algebraEdit

Let   be a finite group, let   be a commutative ring and let   be the group algebra of   over   This algebra is free and a basis can be indexed by the elements of   Most often the basis is identified with  . Every element   can then be uniquely expressed as

  with  .

The multiplication in   extends that in   distributively.

Now let   be a  module and let   be a linear representation of   in   We define   for all   and  . By linear extension   is endowed with the structure of a left- –module. Vice versa we obtain a linear representation of   starting from a  –module  . Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. Therefore, these terms may be used interchangeably.[1][2] This is an example of an isomorphism of categories.

Suppose   In this case the left  –module given by   itself corresponds to the left-regular representation. In the same way   as a right  –module corresponds to the right-regular representation.

In the following we will define the convolution algebra: Let   be a group, the set   is a  –vector space with the operations addition and scalar multiplication then this vector space is isomorphic to   The convolution of two elements   defined by

 

makes   an algebra. The algebra   is called the convolution algebra.

The convolution algebra is free and has a basis indexed by the group elements:   where

 

Using the properties of the convolution we obtain:  

We define a map between   and   by defining   on the basis   and extending it linearly. Obviously the prior map is bijective. A closer inspection of the convolution of two basis elements as shown in the equation above reveals that the multiplication in   corresponds to that in   Thus, the convolution algebra and the group algebra are isomorphic as algebras.

The involution

 

turns   into a  –algebra. We have  

A representation   of a group   extends to a  –algebra homomorphism   by   Since multiplicity is a characteristic property of algebra homomorphisms,   satisfies   If   is unitary, we also obtain   For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see, that, without loss of generality, every linear representation can be assumed to be unitary.

Using the convolution algebra we can implement a Fourier transformation on a group   In the area of harmonic analysis it is shown that the following definition is consistent with the definition of the Fourier transformation on  

Let   be a representation and let   be a  -valued function on  . The Fourier transform   of   is defined as

 

It holds that  

Maps between representationsEdit

A map between two representations   of the same group   is a linear map   with the property that   holds for all   In other words, the following diagram commutes for all  :

 

Such a map is also called  –linear, or an equivariant map. The kernel, the image and the cokernel of   are defined by default. The composition of equivariant maps is again an equivariant map. There is a category of representations with equivariant maps as its morphisms. They are again  –modules. Thus, they provide representations of   due to the correlation described in the previous section.

Irreducible representations and Schur's lemmaEdit

Let   be a linear representation of   Let   be a  –invariant subspace of   i.e.   for all   The restriction   is an isomorphism of   onto itself. Because   holds for all   this construction is a representation of   in   It is called subrepresentation of   Any representation V has at least two subrepresentations, namely the one consisting only of 0, and the one consisting of V itself. The representation is called irreducible representation, if these two are the only subrepresentations. Some authors also call these representations simple, given that they are precisely the simple modules over the group algebra  .

Schur's lemma puts a strong constraint on maps between irreducible representations. If   and   are both irreducible, and   is a linear map such that   for all  , there is the following dichotomy:

  • If   and     is a homothety (i.e.   for a  ). More generally, if   and   are isomorphic, the space of G-linear maps is one-dimensional.
  • Otherwise, if the two representations are not isomorphic, F must be 0.

[3]

PropertiesEdit

Two representations   are called equivalent or isomorphic, if there exists a  –linear vector space isomorphism between the representation spaces. In other words, they are isomorphic if there exists a bijective linear map   such that   for all   In particular, equivalent representations have the same degree.

A representation   is called faithful, if   is injective. In this case   induces an isomorphism between   and the image   As the latter is a subgroup of   we can regard   via   as subgroup of  

We can restrict the range as well as the domain:

Let   be a subgroup of   Let   be a linear representation of   We denote by   the restriction of   to the subgroup  

If there is no danger of confusion, we might use only   or in short  

The notation   or in short   is also used to denote the restriction of the representation   of   onto  

Let   be a function on   We write   or shortly   for the restriction to the subgroup  

It can be proved, that the number of irreducible representations of a group   (or correspondingly the number of simple  –modules) equals the number of conjugacy classes of  

A representation is called semisimple or completely reducible, if it can be written as a direct sum of irreducible representations. This is analogue to the definition of the semisimple algebra.

For the definition of the direct sum of representations please refer to the section on direct sums of representations.

A representation is called isotypic, if it is a direct sum of isomorphic, irreducible representations.

Let   be a given representation of a group   Let   be an irreducible representation of   The  isotype   of   is defined as the sum of all irreducible subrepresentations of   isomorphic to  

Every vector space over   can be provided with an inner product. A representation   of a group   in a vector space endowed with an inner product is called unitary, if   is unitary for every   This means that in particular every   is diagonalizable. For more details see the article on unitary representations.

A representation is unitary with respect to a given inner product if and only if the inner product is invariant with regard to the induced operation of   that means, if   holds for all  

A given inner product   can be replaced by an invariant inner product by exchanging   with

 

Thus, without loss of generality, we can assume that every further considered representation is unitary.

Example. Let   be the dihedral group of order   generated by   which fulfil the properties   and   Let   be a linear representation of   defined on the generators by:

 

This representation is faithful. The subspace   is a  –invariant subspace. Thus, there exists a nontrivial subrepresentation   with   Therefore, the representation is not irreducible. The mentioned subrepresentation is of degree one and irreducible. The complementary subspace of   is  –invariant as well. Therefore, we obtain the subrepresentation   with

 

This subrepresentation is also irreducible. That means, the original representation is completely reducible:

 

Both subrepresentations are isotypic and are the two only non-zero isotypes of  

The representation   is unitary with regard to the standard inner product on   because   and   are unitary.

Let   be any vector space isomorphism. Then   which is defined by the equation   for all   is a representation isomorphic to  

By restricting the domain of the representation to a subgroup, e.g.   we obtain the representation   This representation is defined by the image   whose explicit form is shown above.

ConstructionsEdit

The dual representationEdit

Let   be a given representation. The dual representation or contragredient representation   is a representation of   in the dual vector space of   It is defined by the property

 

With regard to the natural pairing   between   and   the definition above provides the equation:

 

For an example, see the main page on this topic: Dual representation.

Direct sum of representationsEdit

Let   and   be a representation of   and   respectively. The direct sum of these representations is a linear representation and is defined as

 

Let   be representations of the same group   For the sake of simplicity, the direct sum of these representations is defined as a representation of   i.e. it is given as   by viewing   as the diagonal subgroup of  

Example. Let (here   and   are the imaginary unit and the primitive cube root of unity respectively):

 

Then

 

As it is sufficient to consider the image of the generating element, we find, that

 

Tensor product of representationsEdit

Let   be linear representations. We define the linear representation   into the tensor product of   and   by   in which   This representation is called outer tensor product of the representations   and   The existence and uniqueness is a consequence of the properties of the tensor product.

Example. We reexamine the example provided for the direct sum:

 

The outer tensor product

 

Using the standard basis of   we have the following for the generating element:

 

Remark. Note that the direct sum and the tensor products have different degrees and hence are different representations.

Let   be two linear representations of the same group. Let   be an element of   Then   is defined by   for   and we write   Then the map   defines a linear representation of   which is also called tensor product of the given representations.

These two cases have to be strictly distinguished. The first case is a representation of the group product into the tensor product of the corresponding representation spaces. The second case is a representation of the group   into the tensor product of two representation spaces of this one group. But this last case can be viewed as a special case of the first one by focussing on the diagonal subgroup   This definition can be iterated a finite number of times.

Let   and   be representations of the group   Then   is a representation by virtue of the following identity:  . Let   and let   be the representation on   Let   be the representation on   and   the representation on   Then the identity above leads to the following result:

  for all  
Theorem. The irreducible representations of   up to isomorphism are exactly the representations   in which   and   are irreducible representations of   and   respectively.

Symmetric and alternating squareEdit

Let   be a linear representation of   Let   be a basis of   Define   by extending   linearly. It holds that   and therefore   splits up into   in which

 
 

These subspaces are  –invariant and by this define subrepresentations which are called the symmetric square and the alternating square, respectively. These subrepresentations are also defined in   although in this case they are denoted wedge product   and symmetric product   In case that   the vector space   is in general not equal to the direct sum of these two products.

DecompositionsEdit

In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desirable. This can be achieved for finite groups as we will see in the following results. More detailed explanations and proofs may be found in [1] and [2].

Theorem. (Maschke) Let   be a linear representation where   is a vector space over a field of characteristic zero. Let   be a  -invariant subspace of   Then the complement   of   exists in   and is  -invariant.

A subrepresentation and its complement determine a representation uniquely.

The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact – and therefore also of finite – groups:

Theorem. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations.

Or in the language of  -modules: If   the group algebra   is semisimple, i.e. it is the direct sum of simple algebras.

Note that this decomposition is not unique. However, the number of how many times a subrepresentation isomorphic to a given irreducible representation is occurring in this decomposition is independent of the choice of decomposition.

The canonical decomposition

To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition is uniquely determined. It is called the canonical decomposition.

Let   be the set of all irreducible representations of a group   up to isomorphism. Let   be a representation of   and let   be the set of all isotypes of   The projection   corresponding to the canonical decomposition is given by

 

where     and   is the character belonging to  

In the following, we show how to determine the isotype to the trivial representation:

Definition (Projection formula). For every representation   of a group   we define

 

In general,   is not  -linear. We define

 

Then   is a  -linear map, because

 
Proposition. The map   is a projection from   to  

This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.

How often the trivial representation occurs in   is given by   This result is a consequence of the fact that the eigenvalues of a projection are only   or   and that the eigenspace corresponding to the eigenvalue   is the image of the projection. Since the trace of the projection is the sum of all eigenvalues, we obtain the following result

 

in which   denotes the isotype of the trivial representation.

Let   be a nontrivial irreducible representation of   Then the isotype to the trivial representation of   is the null space. That means the following equation holds

 

Let   be a orthonormal basis of   Then we have:

 

Therefore, the following is valid for a nontrivial irreducible representation  :

 

Example. Let   be the permutation groups in three elements. Let   be a linear representation of   defined on the generating elements as follows:

 

This representation can be decomposed on first look into the left-regular representation of  which is denoted by   in the following, and the representation   with

 

With the help of the irreducibility criterion taken from the next chapter, we realize, that   is irreducible and   is not. This is, because for the inner product defined in the section ”Inner product and characters” further below, we have  

The subspace   of   is invariant with respect to the left-regular representation. Restricted to this subspace we obtain the trivial representation.

The orthogonal complement of   is   Restricted to this subspace, which is also  –invariant as we have seen above, we obtain the representation   given by

 

Just like before we can use the irreducibility criterion of the next chapter to prove that   is irreducible. Now,   and   are isomorphic, because   for all   in which   is given by the matrix

 

A decomposition of   in irreducible subrepresentations is:   where   denotes the trivial representation and

 

is the corresponding decomposition of the representation space.

We obtain the canonical decomposition by combining all the isomorphic irreducible subrepresentations:   is the  -isotype of   and consequently the canonical decomposition is given by

 

The theorems above are in general not valid for infinite groups. This will be demonstrated by the following example: let

 

Together with the matrix multiplication   is an infinite group.   acts on   by matrix-vector multiplication. We consider the representation   for all   The subspace   is a  -invariant subspace. However, there exists no  -invariant complement to this subspace. The assumption, that such a complement exists, results in the statement, that every matrix is diagonalizable over   This is known to be wrong and thus presents the contradiction.

That means, if we consider infinite groups, it is possible that a representation, although being not irreducible, can not be decomposed in a direct sum of irreducible subrepresentations.

Character theoryEdit

DefinitionsEdit

The character of a representation   is defined as the map

  in which   denotes the trace of the linear map  [4]

Even though the character is a map between two groups, it is not in general a group homomorphism, as the following example shows.

Let   be the representation defined by:

 

The character   is given by

 

Characters of permutation representations are particularly easy to compute. If V is the G-representation corresponding to the left action of   on a finite set  , then

 

For example,[5] the character of the regular representation   is given by

 

where   denotes the neutral element of  

PropertiesEdit

A crucial property of characters is the formula

 

This formula follows from the fact that the trace of a product AB of two square matrices is the same as the trace of BA. Functions   satisfying such a formula are called class functions. Put differently, class functions and in particular characters are constant on each conjugacy class   It also follows from elementary properties of the trace that   is the sum of the eigenvalues of   with multiplicity. If the degree of the representation is n, then the sum is n long. If s has order m, these eigenvalues are all m-th roots of unity. This fact can be used to show that   and it also implies  

Since the trace of the identity matrix is the number of rows,   where   is the neutral element of   and n is the dimension of the representation. In general,   is a normal subgroup in   The following table shows how the characters   of two given representations   give rise to characters of related representations.

Characters of several standard constructions
Representation Character
dual representation    
direct sum    
tensor product of the representations  

 

symmetric square    
alternating square    

By construction, there is a direct sum decomposition of  . On characters, this corresponds to the fact that the sum of the last two expressions in the table is  , the character of  .

Inner product and charactersEdit

In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups:

Definition (Class functions). A function   is called a class function if it is constant on conjugacy classes of  , i.e.

 

Note that every character is a class function, as the trace of a matrix is preserved under conjugation.

The set of all class functions is a  –algebra and is denoted by  . Its dimension is equal to the number of conjugacy classes of  

Proofs of the following results of this chapter may be found in [1], [2] and [3].

An inner product can be defined on the set of all class functions on a finite group:

 

Orthonormal property. If   are the distinct irreducible characters of  , they form an orthonormal basis for the vector space of all class functions with respect to the inner product defined above, i.e.

  •  
  • Every class function   may be expressed as a unique linear combination of the irreducible characters  .

One might verify that the irreducible characters generate   by showing, that there exists no class function unequal to zero which is orthogonal to all the irreducible characters. To see this, for   a representation and   a class function, denote   Then for   irreducible, we have   from Schur's lemma. Suppose   is a class function which is orthogonal to all the characters. Then by the above we have   whenever   is irreducible. But then it follows that   for all  , by decomposability. Take   to be the regular representation. Applying   to some particular basis element  , we get  . Since this is true for all  , we have  

It follows from the orthonormal property that the number of non-isomorphic irreducible representations of a group   is equal to the number of conjugacy classes of  

Furthermore, a class function on   is a character of   if and only if it can be written as a linear combination of the distinct irreducible characters   with non-negative integer coefficients: if   is a class function on   such that   where   non-negative integers, then   is the character of the direct sum   of the representations   corresponding to   Conversely, it is always possible to write any character as a sum of irreducible characters.

The inner product defined above can be extended on the set of all  -valued functions   on a finite group:

 

Also a symmetric bilinear form can be defined on  

 

These two forms match on the set of characters. If there is no danger of confusion the index of both forms   and   will be omitted.

Let   be two  –modules. Note that  –modules are simply representations of  . Since the orthonormal property yields the number of irreducible representations of   is exactly the number of its conjugacy classes, then there are exactly as many simple  –modules (up to isomorphism) as there are conjugacy classes of  

We define   in which   is the vector space of all  –linear maps. This form is bilinear with respect to the direct sum.

In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations.

For instance, let   and   be the characters of   and   respectively. Then 

Once can deduce the following from the results above, of Schur's lemma and of the complete reducibility of representations.

Theorem. Let   be a linear representation of   with character   Let   where   are irreducible. Let   be an irreducible representation of   with character   Then the number of subrepresentations   which are isomorphic to   is independent of the given decomposition and is equal to the inner product   i.e. the  –isotype   of   is independent of the choice of decomposition. We also get:
 
and thus
 
Corollary. Two representations with the same character are isomorphic. That means, that every representation is determined by its character.

With this we obtain a very handsome result to analyse representations:

Irreducibility criterion. Let   be the character of the representation   then we have   And it holds   if and only if   is irreducible.

Therefore, using the first theorem, the characters of irreducible representations of   form an orthonormal set on   with respect to this inner product.

Corollary. Let   be a vector space with   A given irreducible representation   of   is contained  –times in the regular representation. That means, that if   denotes the regular representation of   we have:   in which   is the set of all irreducible representations of   that are pairwise not isomorphic to each other.

In terms of the group algebra this means, that   as algebras.

As a numerical result we get:

 

in which   is the regular representation and   and   are corresponding characters to   and   respectively. It should also be mentioned, that   denotes the neutral element of the group.

This formula is a necessary and sufficient condition for all irreducible representations of a group up to isomorphism. It provides us with the means to check whether we found all irreducible representations of a group up to isomorphism.

Similarly, by using the character of the regular representation evaluated at   we get the equation:

 

Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations:

The Fourier inversion formula:

 

In addition the Plancherel formula holds:

 

In both formulas   is a linear representation of a group   and  

The corollary above has an additional consequence:

Lemma. Let   be a group. Then the following is equivalent:
  •   is abelian.
  • Every function on   is a class function.
  • All irreducible representations of   have degree  

The induced representationEdit

As was shown in the section on properties of linear representations, we can, by restricting to a subgroup, obtain a representation of a subgroup starting from a representation of a group. Naturally we are interested in the reverse process: Is it possible to obtain the representation of a group starting from a representation of a subgroup? We will see, that the induced representation, which will be defined in the following, provides us with the necessary concept. Admittedly, this construction is not inverse but adjoint to the restriction.

DefinitionsEdit

Let   be a linear representation of   Let   be a subgroup and   the restriction. Let   be a subrepresentation of   We write   to denote this representation. Let   The vector space   depends only on the left coset   of   Let   be a representative system of   then

 

is a subrepresentation of  

A representation   of   in   is called induced by the representation   of   in   if

 

Here   denotes a representative system of   and   for all   and for all   In other words: the representation   is induced by   if every   can be written uniquely as

 

where   for every  

We denote the representation   of   which is induced by the representation   of   as   or in short   if there is no danger of confusion. The representation space itself is frequently used instead of the representation map, i.e.   or   if the representation   is induced by  

Alternative description of the induced representationEdit

By using the group algebra we obtain an alternative description of the induced representation:

Let   be a group,   a  –module and   a  –submodule of   corresponding to the subgroup   of   We say,   is induced by   if   in which   acts on the first factor:   for all  

PropertiesEdit

The results introduced in this section will be presented without proof. These may be found in [1] and [2].

Uniqueness and existence of the induced representation. Let   be a linear representation of a subgroup   of   Then there exists a linear representation   of   which is induced by   Note that this representation is unique up to isomorphism.
Transitivity of induction. Let   be a representation of   and let   be an ascending series of groups. Then we have
 
Lemma. Let   be induced by   and let