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.

Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves 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.

Definition

Linear representations

Let be a –vector space and a finite group. A linear representation of is a group homomorphism Here is notation for a general linear group, and for an automorphism group. This means that 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 use the notation if it is clear to which representation the space belongs.

In this article we will restrict 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

Examples

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, Thus, 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. In other words, 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 representation

Let be a finite set and let be a group acting on Denote by the group of all permutations on with the composition as group multiplication.

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 acts on via The associated linear representation is with for

Left- and right-regular representation

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 algebra

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 multiplicativity 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

This transformation satisfies

Maps between representations

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 lemma

Let be a linear representation of Let be a -invariant subspace of that is, for all and . 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 an 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]

Properties

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 when 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 proven 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 analogous to the corresponding definition for a 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 pairwise 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 i.e. if and only 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.

Constructions

The dual representation

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 representations

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 representations

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 focusing 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 square

Let be a linear representation of Let be a basis of Define by extending linearly. It then 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.

Decompositions

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 an 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 could realize that is irreducible but is not. This is because (in terms of the inner product from ”Inner product and characters” 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

Again, 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 would entail that every matrix is diagonalizable over This is known to be wrong and thus yields a contradiction.

The moral of the story is that if we consider infinite groups, it is possible that a representation - even one that is not irreducible - can not be decomposed into a direct sum of irreducible subrepresentations.

Character theory

Definitions

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

Properties

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 characters

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 nonzero class function which is orthogonal to all the irreducible characters. 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:

A symmetric bilinear form can also 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

It is possible to derive the following theorem from the results above, along with Schur's lemma and 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. This means that every representation is determined by its character.

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

Irreducibility criterion. Let be the character of the representation then we have The case 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. In other words, if denotes the regular representation of then 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. Recall that denotes the neutral element of the group.

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

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 representation

As was shown in the section on properties of linear representations, we can - by restriction - 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 defined below provides us with the necessary concept. Admittedly, this construction is not inverse but rather adjoint to the restriction.

Definitions

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 representation

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 that is induced by if in which acts on the first factor: for all

Properties

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 be a linear representation of Now let be a linear map satisfying the property that for all Then there exists a uniquely determined linear map which extends and for which is valid for all

This means that if we interpret as a –module, we have where is the vector space of all –homomorphisms of to The same is valid for

Induction on class functions. In the same way as it was done with representations, we can - by induction - obtain a class function on the group from a class function on a subgroup. Let be a class function on We define a function on by

We say is induced by and write or

Proposition. The function is a class function on If is the character of a representation of then is the character of the induced representation of
Lemma. If is a class function on and is a class function on then we have:
Theorem. Let be the representation of induced by the representation of the subgroup Let and be the corresponding characters. Let be a representative system of The induced character is given by

Frobenius reciprocity

As a preemptive summary, the lesson to take from Frobenius reciprocity is that the maps and are adjoint to each other.

Let be an irreducible representation of and let be an irreducible representation of then the Frobenius reciprocity tells us that is contained in as often as is contained in

Frobenius reciprocity. If and we have

This statement is also valid for the inner product.

Mackey's irreducibility criterion

George Mackey established a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation.

Two representations and of a group are called disjoint, if they have no irreducible component in common, i.e. if

Let be a group and let be a subgroup. We define for Let be a representation of the subgroup This defines by restriction a representation of We write for We also define another representation of by These two representations are not to be confused.

Mackey's irreducibility criterion. The induced representation is irreducible if and only if the following conditions are satisfied:
  • is irreducible
  • For each the two representations and of are disjoint.[6]

For the case of normal, we have and . Thus we obtain the following:

Corollary. Let be a normal subgroup of Then is irreducible if and only if is irreducible and not isomorphic to the conjugates for

Applications to special groups

In this section we present some applications of the so far presented theory to normal subgroups and to a special group, the semidirect product of a subgroup with an abelian normal subgroup.

Proposition. Let be a normal subgroup of the group and let be an irreducible representation of Then one of the following statements has to be valid:
  • either there exists a proper subgroup of containing , and an irreducible representation of which induces ,
  • or is an isotypic -module.
Proof. Consider as a -module, and decompose it into isotypes as . If this decomposition is trivial, we are in the second case. Otherwise, the larger -action permutes these isotypic modules; because is irreducible as a -module, the permutation action is transitive (in fact primitive). Fix any ; the stabilizer in of is elementarily seen to exhibit the claimed properties.     

Note that if is abelian, then the isotypic modules of are irreducible, of degree one, and all homotheties.

We obtain also the following

Corollary. Let be an abelian normal subgroup of and let be any irreducible representation of We denote with the index of in Then [1]

If is an abelian subgroup of (not necessarily normal), generally is not satisfied, but nevertheless is still valid.

Classification of representations of a semidirect product

In the following, let be a semidirect product such that the normal semidirect factor, , is abelian. The irreducible representations of such a group can be classified by showing that all irreducible representations of can be constructed from certain subgroups of . This is the so-called method of “little groups” of Wigner and Mackey.

Since is abelian, the irreducible characters of have degree one and form the group The group acts on by for

Let be a representative system of the orbit of in For every let This is a subgroup of Let be the corresponding subgroup of We now extend the function onto by for Thus, is a class function on Moreover, since for all it can be shown that is a group homomorphism from to Therefore, we have a representation of of degree one which is equal to its own character.

Let now be an irreducible representation of Then we obtain an irreducible representation of by combining with the canonical projection Finally, we construct the tensor product of and Thus, we obtain an irreducible representation of

To finally obtain the classification of the irreducible representations of we use the representation of which is induced by the tensor product Thus, we achieve the following result:

Proposition.
  • is irreducible.
  • If and are isomorphic, then and additionally is isomorphic to
  • Every irreducible representation of is isomorphic to one of the

Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in [1].

In other words, we classified all irreducible representations of

Representation ring

The representation ring of is defined as the abelian group

With the multiplication provided by the tensor product, becomes a ring. The elements of are called virtual representations.

The character defines a ring homomorphism in the set of all class functions on with complex values

in which the are the irreducible characters corresponding to the

Because a representation is determined by its character, is injective. The images of are called virtual characters.

As the irreducible characters form an orthonormal basis of induces an isomorphism

This isomorphism is defined on a basis out of elementary tensors by respectively and extended bilinearly.

We write for the set of all characters of and to denote the group generated by i.e. the set of all differences of two characters. It then holds that and Thus, we have and the virtual characters correspond to the virtual representations in an optimal manner.

Since holds, is the set of all virtual characters. As the product of two characters provides another character, is a subring of the ring of all class functions on Because the form a basis of we obtain, just as in the case of an isomorphism

Let be a subgroup of The restriction thus defines a ring homomorphism which will be denoted by or Likewise, the induction on class functions defines a homomorphism of abelian groups which will be written as or in short

According to the Frobenius reciprocity, these two homomorphisms are adjoint with respect to the bilinear forms and Furthermore, the formula shows that the image of is an ideal of the ring

By the restriction of representations, the map can be defined analogously for and by the induction we obtain the map for Due to the Frobenius reciprocity, we get the result that these maps are adjoint to each other and that the image is an ideal of the ring

If is a commutative ring, the homomorphisms and may be extended to –linear maps:

in which are all the irreducible representations of up to isomorphism.

With we obtain in particular that and supply homomorphisms between and

Let and be two groups with respective representations and Then, is the representation of the direct product as was shown in a previous section. Another result of that section was that all irreducible representations of are exactly the representations where and are irreducible representations of and respectively. This passes over to the representation ring as the identity in which is the tensor product of the representation rings as –modules.

Induction theorems

Induction theorems relate the representation ring of a given finite group G to representation rings of a family X consisting of some subsets H of G. More precisely, for such a collection of subgroups, the induction functor yields a map

; induction theorems give criteria for the surjectivity of this map or closely related ones.

Artin's induction theorem is the most elementary theorem in this group of results. It asserts that the following are equivalent:

  • The cokernel of is finite.
  • is the union of the conjugates of the subgroups belonging to i.e.

Since is finitely generated as a group, the first point can be rephrased as follows:

  • For each character of there exist virtual characters and an integer such that

Serre (1977) gives two proofs of this theorem. For example, since G is the union of its cyclic subgroups, every character of is a linear combination with rational coefficients of characters induced by characters of cyclic subgroups of Since the representations of cyclic groups are well-understood, in particular the irreducible representations are one-dimensional, this gives a certain control over representations of G.

Under the above circumstances, it is not in general true that is surjective. Brauer's induction theorem asserts that is surjective, provided that X is the family of all elementary subgroups. Here a group H is elementary if there is some prime p such that H is the direct product of a cyclic group of order prime to and a –group. In other words, every character of is a linear combination with integer coefficients of characters induced by characters of elementary subgroups. The elementary subgroups H arising in Brauer's theorem have a richer representation theory than cyclic groups, they at least have the property that any irreducible representation for such H is induced by a one-dimensional representation of a (necessarily also elementary) subgroup . (This latter property can be shown to hold for any supersolvable group, which includes nilpotent groups and, in particular, elementary groups.) This ability to induce representations from degree 1 representations has some further consequences in the representation theory of finite groups.

Real representations

For proofs and more information about representations over general subfields of please refer to [2].

If a group acts on a real vector space the corresponding representation on the complex vector space is called real ( is called the complexification of ). The corresponding representation mentioned above is given by for all

Let be a real representation. The linear map is -valued for all Thus, we can conclude that the character of a real representation is always real-valued. But not every representation with a real-valued character is real. To make this clear, let be a finite, non-abelian subgroup of the group

Then acts on Since the trace of any matrix in is real, the character of the representation is real-valued. Suppose is a real representation, then would consist only of real-valued matrices. Thus, However the circle group is abelian but was chosen to be a non-abelian group. Now we only need to prove the existence of a non-abelian, finite subgroup of To find such a group, observe that can be identified with the units of the quaternions. Now let The following two-dimensional representation of is not real-valued, but has a real-valued character:

Then the image of is not real-valued, but nevertheless it is a subset of Thus, the character of the representation is real.

Lemma. An irreducible representation of is real if and only if there exists a nondegenerate symmetric bilinear form on preserved by

An irreducible representation of on a real vector space can become reducible when extending the field to For example, the following real representation of the cyclic group is reducible when considered over

Therefore, by classifying all the irreducible representations that are real over we still haven't classified all the irreducible real representations. But we achieve the following:

Let be a real vector space. Let act irreducibly on and let If is not irreducible, there are exactly two irreducible factors which are complex conjugate representations of

Definition. A quaternionic representation is a (complex) representation which possesses a –invariant anti-linear homomorphism satisfying Thus, a skew-symmetric, nondegenerate –invariant bilinear form defines a quaternionic structure on

Theorem. An irreducible representation is one and only one of the following:
(i) complex: is not real-valued and there exists no –invariant nondegenerate bilinear form on
(ii) real: a real representation; has a –invariant nondegenerate symmetric bilinear form.
(iii) quaternionic: is real, but is not real; has a –invariant skew-symmetric nondegenerate bilinear form.

Representations of particular groups

Symmetric groups

Representation of the symmetric groups have been intensely studied. Conjugacy classes in (and therefore, by the above, irreducible representations) correspond to partitions of n. For example, has three irreducible representations, corresponding to the partitions

3; 2+1; 1+1+1

of 3. For such a partition, a Young tableau is a graphical device depicting a partition. The irreducible representation corresponding to such a partition (or Young tableau) is called a Specht module.

Representations of different symmetric groups are related: any representation of yields a representation of by induction, and vice versa by restriction. The direct sum of all these representation rings

inherits from these constructions the structure of a Hopf algebra which, it turns out, is closely related to symmetric functions.

Finite groups of Lie type

To a certain extent, the representations of the , as n varies, have a similar flavor as for the ; the above-mentioned induction process gets replaced by so-called parabolic induction. However, unlike for , where all representations can be obtained by induction of trivial representations, this is not true for . Instead, new building blocks, known as cuspidal representations, are needed.

Representations of and more generally, representations of finite groups of Lie type have been thoroughly studied. Bonnafé (2010) describes the representations of . A geometric description of irreducible representations of such groups, including the above-mentioned cuspidal representations, is obtained by Deligne-Lusztig theory, which constructs such representation in the l-adic cohomology of Deligne-Lusztig varieties.

The similarity of the representation theory of and goes beyond finite groups. The philosophy of cusp forms highlights the kinship of representation theoretic aspects of these types of groups with general linear groups of local fields such as Qp and of the ring of adeles, see Bump (2004).

Outlook—Representations of compact groups

The theory of representations of compact groups may be, to some degree, extended to locally compact groups. The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms. For proofs, further information and for a more detailed insight which is beyond the scope of this chapter please consult [4] and [5].

Definition and properties

A topological group is a group together with a topology with respect to which the group composition and the inversion are continuous. Such a group is called compact, if any cover of which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again.

Let be a compact group and let be a finite-dimensional –vector space. A linear representation of to is a continuous group homomorphism i.e. is a continuous function in the two variables and

A linear representation of into a Banach space is defined to be a continuous group homomorphism of into the set of all bijective bounded linear operators on with a continuous inverse. Since we can do without the last requirement. In the following, we will consider in particular representations of compact groups in Hilbert spaces.

Just as with finite groups, we can define the group algebra and the convolution algebra. However, the group algebra provides no helpful information in the case of infinite groups, because the continuity condition gets lost during the construction. Instead the convolution algebra takes its place.

Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart to the summation over a finite group:

Existence and uniqueness of the Haar measure

On a compact group there exists exactly one measure such that:

  • It is a left-translation-invariant measure
  • The whole group has unit measure:

Such a left-translation-invariant, normed measure is called Haar measure of the group

Since is compact, it is possible to show that this measure is also right-translation-invariant, i.e. it also applies

By the scaling above the Haar measure on a finite group is given by for all

All the definitions to representations of finite groups that are mentioned in the section ”Properties”, also apply to representations of compact groups. But there are some modifications needed:

To define a subrepresentation we now need a closed subspace. This was not necessary for finite-dimensional representation spaces, because in this case every subspace is already closed. Furthermore, two representations of a compact group are called equivalent, if there exists a bijective, continuous, linear operator between the representation spaces whose inverse is also continuous and which satisfies for all

If is unitary, the two representations are called unitary equivalent.

To obtain a –invariant inner product from a not –invariant, we now have to use the integral over instead of the sum. If is an inner product on a Hilbert space which is not invariant with respect to the representation of then

is a –invariant inner product on due to the properties of the Haar measure Thus, we can assume every representation on a Hilbert space to be unitary.

Let be a compact group and let Let be the Hilbert space of the square integrable functions on We define the operator on this space by where

The map is a unitary representation of It is called left-regular representation. The right-regular representation is defined similarly. As the Haar measure of is also right-translation-invariant, the operator on is given by The right-regular representation is then the unitary representation given by The two representations and are dual to each other.

If is infinite, these representations have no finite degree. The left- and right-regular representation as defined at the beginning are isomorphic to the left- and right-regular representation as defined above, if the group is finite. This is due to the fact that in this case

Constructions and decompositions

The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. The direct sum and the tensor product with a finite number of summands/factors are defined in exactly the same way as for finite groups. This is also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem saying that the irreducible representations of the product of two groups are (up to isomorphism) exactly the tensor product of the irreducible representations of the factor groups. First, we note that the direct product of two compact groups is again a compact group when provided with the product topology. The Haar measure on the direct product is then given by the product of the Haar measures on the factor groups.

For the dual representation on compact groups we require the topological dual of the vector space This is the vector space of all continuous linear functionals from the vector space into the base field. Let be a representation of a compact group in

The dual representation is defined by the property

Thus, we can conclude that the dual representation is given by for all The map is again a continuous group homomorphism and thus a representation.

On Hilbert spaces: is irreducible if and only if is irreducible.

By transferring the results of the section decompositions to compact groups, we obtain the following theorems:

Theorem. Every irreducible representation of a compact group into a Hilbert space is finite-dimensional and there exists an inner product on such that is unitary. Since the Haar measure is normalized, this inner product is unique.

Every representation of a compact group is isomorphic to a direct Hilbert sum of irreducible representations.

Let be a unitary representation of the compact group Just as for finite groups we define for an irreducible representation the isotype or isotypic component in to be the subspace

This is the sum of all invariant closed subspaces which are –isomorphic to

Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal.

Theorem.
(i) is a closed invariant subspace of
(ii) is –isomorphic to the direct sum of copies of
(iii) Canonical decomposition: is the direct Hilbert sum of the isotypes in which passes through all the isomorphism classes of the irreducible representations.

The corresponding projection to the canonical decomposition in which is an isotype of is for compact groups given by

where and is the character corresponding to the irreducible representation

Projection formula

For every representation of a compact group we define

In general is not –linear. Let

The map is defined as endomorphism on by having the property

which is valid for the inner product of the Hilbert space

Then is –linear, because of

where we used the invariance of the Haar measure.

Proposition. The map is a projection from to

If the representation is finite-dimensional, it is possible to determine the direct sum of the trivial subrepresentation just as in the case of finite groups.

Characters, Schur's lemma and the inner product

Generally, representations of compact groups are investigated on Hilbert- and Banach spaces. In most cases they are not finite-dimensional. Therefore, it is not useful to refer to characters when speaking about representations of compact groups. Nevertheless, in most cases it is possible to restrict the study to the case of finite dimensions:

Since irreducible representations of compact groups are finite-dimensional and unitary (see results from the first subsection), we can define irreducible characters in the same way as it was done for finite groups.

As long as the constructed representations stay finite-dimensional, the characters of the newly constructed representations may be obtained in the same way as for finite groups.

Schur's lemma is also valid for compact groups:

Let be an irreducible unitary representation of a compact group Then every bounded operator satisfying the property for all is a scalar multiple of the identity, i.e. there exists such that

Definition. The formula

defines an inner product on the set of all square integrable functions of a compact group Likewise

defines a bilinear form on of a compact group

The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid:

Theorem. Let and be the characters of two non-isomorphic irreducible representations and respectively. Then the following is valid
  • i.e. has "norm"
Theorem. Let be a representation of with character Suppose is an irreducible representation of with character The number of subrepresentations of equivalent to is independent of any given decomposition for and is equal to the inner product
Irreducibility Criterion. Let be the character of the representation then is a positive integer. Moreover 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. Every irreducible representation of is contained –times in the left-regular representation.
Lemma. Let be a compact group. Then the following statements are equivalent:
  • is abelian.
  • All the irreducible representations of have degree
Orthonormal Property. Let be a group. The non-isomorphic irreducible representations of form an orthonormal basis in with respect to this inner product.

As we already know that the non-isomorphic irreducible representations are orthonormal, we only need to verify that they generate This may be done, by proving that there exists no non-zero square integrable function on orthogonal to all the irreducible characters.

Just as in the case of finite groups, the number of the irreducible representations up to isomorphism of a group equals the number of conjugacy classes of However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information.

The induced representation

If is a closed subgroup of finite index in a compact group the definition of the induced representation for finite groups may be adopted.

However, the induced representation can be defined more generally, so that the definition is valid independent of the index of the subgroup

For this purpose let be a unitary representation of the closed subgroup The continuous induced representation is defined as follows:

Let denote the Hilbert space of all measurable, square integrable functions with the property for all The norm is given by

and the representation is given as the right-translation:

The induced representation is then again a unitary representation.

Since is compact, the induced representation can be decomposed into the direct sum of irreducible representations of Note that all irreducible representations belonging to the same isotype appear with a multiplicity equal to

Let be a representation of then there exists a canonical isomorphism

The Frobenius reciprocity transfers, together with the modified definitions of the inner product and of the bilinear form, to compact groups. The theorem now holds for square integrable functions on instead of class functions, but the subgroup must be closed.

The Peter-Weyl Theorem

Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements.

The Peter-Weyl Theorem. Let be a compact group. For every irreducible representation of let be an orthonormal basis of We define the matrix coefficients for Then we have the following orthonormal basis of :

We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups:

The Peter-Weyl Theorem (Second version).[7] There exists a natural –isomorphism
in which is the set of all irreducible representations of up to isomorphism and is the representation space corresponding to More concretely:

History

The general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. Later the modular representation theory of Richard Brauer was developed.

See also

Literature

  • Bonnafé, Cedric (2010). Representations of SL2(Fq). Algebra and Applications. Vol. 13. Springer. ISBN 9780857291578.
  • Bump, Daniel (2004), Lie Groups, Graduate Texts in Mathematics, vol. 225, New York: Springer-Verlag, ISBN 0-387-21154-3
  • [1] Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, New York: Springer Verlag, ISBN 0-387-90190-6
  • [2] Fulton, William; Harris, Joe: Representation Theory A First Course. Springer-Verlag, New York 1991, ISBN 0-387-97527-6.
  • [3] Alperin, J.L.; Bell, Rowen B.: Groups and Representations Springer-Verlag, New York 1995, ISBN 0-387-94525-3.
  • [4] Deitmar, Anton: Automorphe Formen Springer-Verlag 2010, ISBN 978-3-642-12389-4, p. 89-93,185-189
  • [5] Echterhoff, Siegfried; Deitmar, Anton: Principles of harmonic analysis Springer-Verlag 2009, ISBN 978-0-387-85468-7, p. 127-150
  • [6] Lang, Serge: Algebra Springer-Verlag, New York 2002, ISBN 0-387-95385-X, p. 663-729
  • [7] Sengupta, Ambar (2012). Representing finite groups: a semisimple introduction. New York. ISBN 9781461412311. OCLC 769756134.{{cite book}}: CS1 maint: location missing publisher (link)

References

  1. ^ (Serre 1977, p. 47)
  2. ^ (Sengupta 2012, p. 62)
  3. ^ Proof. Suppose is nonzero. Then is valid for all Therefore, we obtain for all and And we know now, that is –invariant. Since is irreducible and we conclude Now let This means, there exists such that and we have Thus, we deduce, that is a –invariant subspace. Because is nonzero and is irreducible, we have Therefore, is an isomorphism and the first statement is proven. Suppose now that Since our base field is we know that has at least one eigenvalue Let then and we have for all According to the considerations above this is only possible, if i.e.
  4. ^ Some authors define the character as , but this definition is not used in this article.
  5. ^ by using the action of G on itself given by
  6. ^ A proof of this theorem may be found in [1].
  7. ^ A proof of this theorem and more information regarding the representation theory of compact groups may be found in [5].

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