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.
With the exception of 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.
Let be a –vector space and a finite group. A linear representation of a finite group 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
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
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. 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.
turns into a –algebra. We have
A representation of a group extends to a –algebra homomorphism by Since multiplicity is a characteristic property of algebra homomorphisms, satisfies If is unitary, we also obtain For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary.
Using the convolution algebra we can implement a Fourier transformation on a group In the area of harmonic analysis it is shown that the following definition is consistent with the definition of the Fourier transformation on
Let be a representation and let be a -valued function on . The Fourier transform of is defined as
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.
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.
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.
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):
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.
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  and .
- 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.
The character of a representation is defined as the map
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
where denotes the neutral element of
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.
|tensor product of the representations||
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
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.
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
- 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
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.
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 an -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 
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.
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:
- 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 .
In other words, 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