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.
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 Also the notation is sometimes used to denote the degree of a representation
A very elementary example is the trivial representation, which is given by for all
A representation of degree of a group is a homomorphism in the multiplicative group As every element of is of finite order, the values of are roots of unity.
Another nontrivial example:
Let be a nontrivial linear representation.
As generates the group , the group homomorphism is determined by its value on Because is a group homomorphism, it has to satisfy And as is nontrivial, we conclude 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:
A third example:
Let be a group and let be the group homomorphism defined by:
In this case is a linear representation of of degree
Let be a finite set. Let be a group operating on
The group is consequently 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 from to given by for all All linear maps are uniquely defined by this property.
Example
Let be and Then operates on via
The associated linear representation is with for
Let be a group of order Let 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 similarly. As before we use a vector space with And as before we choose again a basis of and index it with the elements of The right-regular representation is defined by 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 be equal to and let the representation space be Let be a basis of
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 leftmodule.
Vice versa we obtain a linear representation of starting from a module Therefore, these terms may be used interchangeably.
Suppose In this case the leftmodule given by itself corresponds to the left-regular representation. In the same way as a rightmodule 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. Let 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 complex-valued function on . The Fourier transform of is defined as
A map between two representations of the same group is a linear map with the property that holds for all
Such a map is also called linear.
The kernel, the image and the cokernel of are defined by default. They are again modules. Thus, they provide representations of due to the correlation described in the previous section.
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
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
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
A representation is called irreducible or simple, if there are no nontrivial invariant vector subspaces of Here, as well as in the following text, we include the whole vector space as well as the zero-vector space in our definition of trivial vector subspaces.
In terms of the group algebra the irreducible representations correspond to the simplemodules.
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.
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
for all and
With regard to the natural pairing between and the definition above provides the equation:
Let and be a representation of and respectively.
The direct sum of these representations is defined as in which for all and
In this manner becomes a linear representation.
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 be the linear representation given by
And let the linear representation given by
Then is a linear representation of in
Using the standard basis and the matrix form of the representation, the direct sum takes the following form:
As it is sufficient to consider the image of the generating element, we find, that :
is given by:
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.
Let and 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.
This result enables us to study the representations of by studying the single representations of and
Example
We pick up again the example of the direct sum to show the difference between the direct sum and the tensor product.
Let be the linear representation given by
And let be the linear representation given by
Then the outer tensor product is given by in which
The linear map belonging to the generating element is given by:
where we used the standard basis of
A comparison with the direct sum reveals the difference. The representations obtained in this way do not even have the same degree.
Symmetric and alternating square
Let be a linear representation of Let be a basis of Define by extending linearly. It holds and Because of that splits up into in which
and
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.
In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desireable.
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
This theorem is valid for vector spaces over a field of characteristic zero:
Let be a linear representation and 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
This theorem applies for representations over fields of characteristic zero:
Every linear representation of compact groups is a direct sum of irreducible representations.
This means in the phrasing 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:
Projection formula
Let be a group with
For every representation of we define
In general, is not linear.
We define
Then is a linear map, because for all
This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.
The number how often the trivial representation occurs in is given by the trace of
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 and where
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
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 in the following example:
Let Together with the matrix multiplication is a group of infinite cardinality. The group is acting 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.
Let be a linear representation of a finite group into the vector space We define the map by in which denotes the trace of the linear map The complex-valued function obtained by this is called character of the representation
Sometimes the character of a representation is defined as in which denotes the degree of the representation. In this article this definition is not used.
Based on the definition above it is obvious, that isomorphic representations have the same character.
Examples
An elementary example is a representation of degree one. Its character is given by
As another example we consider the permutation representation of corresponding to the left action of on a finite set In this case the character is given by
where denotes the neutral element of
Note, that in this context it is correct to use the notion of regular representation and not to distinguish between left- and right-regular as they are isomorphic and thus have the same character.
As a last example we consider the group Let be the representation defined by:
The character is given by
This example shows, that the character is in general no group homomorphism.
As shown in the section on properties of linear representations every representation can be assumed to be unitary.
A character is called unitary, if it belongs to a unitary representation.
A character is called irreducible, if the corresponding representation is irreducible.
Let be the character of a (unitary) representation of degree Then the following holds:
where is the neutral element of
is the sum of the eigenvalues of with multiplicity.
The character of the representation belonging to is given by
Let be the character of and the character of Then the character of is given by
Let be a linear representation of and let be the corresponding character. Let be the character of the symmetric square and let be the character of the alternating square. For every the following holds:
Let and be two irreducible representations. Let be a linear map such that for all Then the following is valid:
If and are not isomorphic, we have
If and is a homothety (i.e. for a ).
Proof
Suppose 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 a such that and we have Thus, we deduce, that is a invariant subspace. Because 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.
In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups:
The Class functions:
A function on which fulfils the equation is called class function.
The set of all class functions is a Algebra. The dimension of is equal to the number of conjugacy classes of
Theorem
Let be the distinct irreducible characters of
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 coefficients.
Proof
Let be such that where for all Consequently, is the character of the direct sum of the representations corresponding to the . Conversely, it is always possible to write any character as a sum of irreducible characters.
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 complex-valued functions on a finite group:
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. 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.
Theorem
Let and be the characters of two non-isomorphic irreducible representations and respectively. Then the following is valid
i.e. has „norm“
Corollary
Let and be the characters of and respectively. Then the following holds:
This corollary is a direct consequence of the theorem 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 it holds:
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:
Finally, we recall the definition of class functions in order to value the exceptional position of characters within them:
Orthonormal property
Let be a group. The non-isomorphic irreducible characters of form an orthonormal basis of with regard to the inner product defined at the beginning of this section.
I.e. for two irreducible characters and the following is valid:
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.
Equivalent to the orthonormal property we have:
The number of non-isomorphic irreducible representations of a group is equal to the number of conjugacy classes of
In terms of the group algebra this means, that there are exactly as many simple modules (up to isomorphism) as there are conjugacy classes of
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.
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. In mathematical notations, the representation spaces are also frequently used instead of the representation mappings, i.e. or in short if the representation V is induced by W.
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, is induced by if in which acts on the first factor: for all
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
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, using the so-called induction, obtain a class function on the group out of a class function on a subgroup.
Let be a class function on We define the 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 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
The message of the 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
Proof
Every class function can be written as a linear combination of irreducible characters. As is a bilinear form, we can, without loss of generality, assume and to be characters of irreducible representations of in and of in respectively.
We define for all
Then we have
In the course of this sequence of equations we used only the definition of induction on class functions and the properties of characters.
Alternative proof
In terms of the group algebra, i.e. by the alternative description of the induced representation, the Frobenius reciprocity is a special case of a general equation for a change of rings:
This equation is by definition equivalent to
As this bilinear form tallies the bilinear form on the corresponding characters, the theorem follows without calculation.
George Mackey has 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.
Starting from this theorem we obtain directly 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
Proof
As is normal, we have and Thus, the statement follows directly from the criterion of Mackey.
In this chapter 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 true subgroup of which contains and an irreducible representation of which induces
or the restriction of onto is isotypic.
If is abelian, the second point of the proposition above is equivalent to the statement, that is a homothety for every
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
If is an abelian subgroup of (not necessarily normal), generally is not satisfied, but nevertheless is still valid.
In the following, let and be subgroups of the group where is assumed to be normal and abelian. Additionally, assume that is the semidirect product of and i.e. .
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 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].
I.e. we classified all irreducible representations of
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
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 holds
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.
Theorem
Let be a family of subgroups of a finite group Let be the homomorphism defined by the family of the Then the following properties are equivalent:
A group is called elementary, if it is the direct product of a cyclic group of prime order and a group.
A subgroup of is called elementary, if it is elementary for at least one prime number
A representation of is called monomial, if it is induced by a degreerepresentation of a subgroup of
Brauer's Theorem
Every character of is a linear combination with integer coefficients of characters induced by characters of elementary subgroups.
A proof and a detailed explanation to Brauer's theory may be found in [1] and [6].
Since elementary groups are nilpotent and thus supersolvable, the following theorem taken from [1] can be applied:
Theorem
Let be a supersolvable group. Then every irreducible representation of is induced by a representation of degree of a subgroup of I.e. every irreducible representation of is monomial.
Thus, we achieve the following result of Brauer's theorem:
Theorem
Every character of is a linear combination with integer coefficients of monomial characters.
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 vector space is called real.
The vector space is a complex vector space also called complexification of
The corresponding representation mentioned above is given by for all
Let be a real representation. The linear map is real-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 a matrix in is real, the character of the representation is real-valued.
Suppose would be a real representation, then would consist only of real-valued matrices. Thus, would be a subgroup of the circle group
The circle group is abelian and so are all its subgroups, especially But was chosen to be a non-abelian group.
Now we only need to prove the existence of such a non-abelian, finite subgroup of To find such a group, observe that can be identified with the units of the quaternions. Now let
We will now give an example of a two-dimensional representation of which is not real-valued, but has a real-valued character:
Let be a group homomorphism determined by:
Then the image of is not real-valued, but nevertheless it is a subset of Thus, the character of the representation is real.
An irreducible representation of on a vector space with base field can become reducible when extending the field to
An example is the irreducible representation of the cyclic group in given by
which is reducible when considered over
I.e. 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 be the corresponding real representation of
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:
complex: is not real-valued and there exists no invariant nondegenerate bilinear form on
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].
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
On a compact group there exists exactly one measure such that:
for all i.e. the measure is left-translation-invariant.
thus the whole group has 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
for all
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
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.
Additionally, we need a Haar measure on the direct product of groups in order to obtain also for compact groups the result, that the irreducible representations of the product of two groups are, up to isomorphism, exactly the tensor product of the irreducible representations of the single groups.
The direct product of two compact groups is again a compact group, if provided with the product topology.
The Haar measure on this group is given by the product of the Haar measures of the single 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 for all
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
is a closed invariant subspace of
is isomorphic to the direct sum of copies of
is the direct Hilbert sum of the isotypes in which passes through all the isomorphism classes of the irreducible representations. This decomposition of is called the canonical decomposition.
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 be
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.
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
Defintions
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.
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 Let in which the are irreducible. As the direct sum is finite, the sum of the irreducible characters corresponding to the defines a character of Let now be an irreducible representation of with character
Then we have:
The number of subrepresentations equivalent to are independent of the given decomposition and is equal to the inner product
I.e. the isotype of is independent of the choice of the decomposition and it holds:
Theorem
Two irreducible representations with the same character are isomorphic.
Irreducibility criterion
Let be the character of the representation then Additionally 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 we note:
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.
If is a closed subgroup of finite index in the 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 and the subgroup must be closed.
Another important result in the representation theory of compact groups is the Theorem of Peter-Weyl. It is usually presented and proved in the harmonic analysis, as it represents one of its central and fundamental statements.
Theorem of Peter-Weyl
Let be a compact group. For every irreducible representation of let be an orthonormal basis of
Second version of the Theorem of Peter-Weyl
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
This isomorphism maps a given onto where
Thus, we have a generalisation of the Fourier series for functions on compact groups.
Actually, this theorem is just a reformulation of the first version.
A proof of this theorem and more information regarding the representation theory of compact groups may be found in [5].