Wikipedia / Representation theory of finite groups
Wikipedia / 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
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
a finite group. A linear representation is a group homomorphism is notation for a general linear group for an automorphism group is called a representation space
The representation of a group in a module instead of a vector space is also called a linear representation.
belongs.
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
Examples
The trivial representation
is a homomorphism into the multiplicative group are roots of unity has to be one of the following three maps:
be the group homomorphism defined by:
Permutation representation
be a finite set with the composition as group multiplication.
are uniquely defined by this property.
Example.
Left- and right-regular representation
The left-regular representation is a special case of the permutation representation The degree of the left-regular representation is equal to the order of the group.
The right-regular representation
These two representations are isomorphic For this reason they are not always set apart, and often referred to as "the" regular representation.
is isomorphic to the left-regular representation if and only if
Example.
Representations, modules and the convolution algebra
be a commutative ring be the group algebra can then be uniquely expressed as
.
distributively.
–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.
–module corresponds to the right-regular representation.
In the following we will define the convolution algebra–vector space with the operations addition and scalar multiplication defined by
an algebra is called the convolution algebra.
where
and extending it linearly. Obviously the prior map is bijective Thus, the convolution algebra and the group algebra are isomorphic as algebras.
The involution
into a ^*–algebra
For the definition of a unitary representation, please refer to the section on properties. In that section 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 In the area of harmonic analysis
is defined as
Maps between representations
:
Such a map is also called –linear, or an equivariant map. The kernel, the image and the cokernel 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 due to the correlation described in the previous section.
Irreducible representations and Schur's lemma
It is called subrepresentation 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.
Schur's lemma, there is the following dichotomy:
is a homothety are isomorphic, the space of G-linear maps is one-dimensional.
Otherwise, if the two representations are not isomorphic, F must be 0.
Properties
are called equivalent or isomorphic In particular, equivalent representations have the same degree.
is called faithful is injective
We can restrict the range as well as the domain:
–modules) equals the number of conjugacy classes
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.
–isotype
can be provided with an inner product in a vector space endowed with an inner product is called unitary is unitary is diagonalizable. For more details see the article on unitary representations.
with
Thus, without loss of generality we can assume that every further considered representation is unitary.
Example. be the dihedral group of order defined on the generators by:
Therefore, the representation is not irreducible. The mentioned subrepresentation is of degree one and irreducible. The complementary subspace with
This subrepresentation is also irreducible. That means, the original representation is completely reducible:
are unitary.
whose explicit form is shown above.
Constructions
The dual representation
be a given representation. The dual representation or contragredient representation in the dual vector space It is defined by the property
the definition above provides the equation:
For an example, see the main page on this topic: Dual representation.
Direct sum of representations
respectively. The direct sum of these representations is a linear representation and is defined as
Example. 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
into the tensor product This representation is called outer tensor product 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
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.
which is also called tensor product of the given representations.
This definition can be iterated a finite number of times.
Then the identity above leads to the following result:
Theorem. respectively.
Symmetric and alternating square
in which
–invariant and by this define subrepresentations which are called the symmetric square and the alternating square 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 and .
Theorem. (Maschke)-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.
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.
The projection corresponding to the canonical decomposition is given by
In the following, we show how to determine the isotype to the trivial representation:
Definition (Projection formula). we define
-linear. We define
-linear map, because
Proposition. is a projection
This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.
This result is a consequence of the fact that the eigenvalues of a projection is the image of the projection. Since the trace of the projection is the sum of all eigenvalues, we obtain the following result
denotes the isotype of the trivial representation.
is the null space. That means the following equation holds
be an orthonormal basis Then we have:
:
Example. defined on the generating elements as follows:
with
With the help of the irreducibility criterion is not. This is because (in terms of the inner product from ”Inner product and characters”
is invariant with respect to the left-regular representation. Restricted to this subspace we obtain the trivial representation.
given by
is given by the matrix
denotes the trivial representation and
is the corresponding decomposition of the representation space.
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
-invariant complement to this subspace. The assumption that such a complement exists would entail that every matrix is diagonalizable 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 is defined as the map
denotes the trace
Even though the character is a map between two groups, it is not in general a group homomorphism, as the following example shows.
be the representation defined by:
is given by
Characters of permutation representations are particularly easy to compute. If V is the G, then
For example, the character of the regular representation is given by
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 satisfying such a formula are called class functions. Put differently, class functions and in particular characters are constant on each conjugacy class is the sum of the eigenvalues 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
Since the trace of the identity matrix and n is a normal subgroup give rise to characters of related representations..
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). is called a class function, i.e.
Note that every character is a class function, as the trace of a matrix is preserved under conjugation.
Proofs of the following results of this section may be found in , , and .
An inner product can be defined on the set of all class functions on a finite group:
Orthonormal property., they form an orthonormal basis for the vector space of all class functions with respect to the inner product defined above, i.e.
.
from Schur's lemma
is equal to the number of conjugacy classes
Conversely, it is always possible to write any character as a sum of irreducible characters.
The inner product on a finite group:
will be omitted.
–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.
It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations.
Theorem. 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. is irreducible.
form an orthonormal set with respect to this inner product.
Corollary.–times in the regular representation that are pairwise not isomorphic to each other.
as algebras.
As a numerical result we get:
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.
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:
The corollary above has an additional consequence:
Lemma. be a group. Then the following is equivalent:
is abelian.
is a class function.
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
depends only on the left coset be a representative system then
is called induced if
can be written uniquely as
Alternative description of the induced representation
By using the group algebra we obtain an alternative description of the induced representation:
Properties
The results introduced in this section will be presented without proof. These may be found in and .
Uniqueness and existence of the induced representation. Note that this representation is unique up to isomorphism.
Transitivity of induction. be an ascending series of groups. Then we have
Lemma.
Induction on class functions. In the same way as it was done with representations, we can—by induction by
is induced
Proposition. is the character
Lemma.
Theorem. The induced character is given by
Frobenius reciprocity
are adjoint to each other.
Frobenius reciprocity.
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.
are called disjoint
These two representations are not to be confused.
Mackey's irreducibility criterion. is irreducible if and only if the following conditions are satisfied:
is irreducible
are disjoint. A proof of this theorem may be found in .
. Thus we obtain the following:
Corollary.
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. be a normal subgroup Then one of the following statements has to be valid:
,
-module.
Proof.-module, the permutation action is transitive (in fact primitive; the stabilizer is elementarily seen to exhibit the claimed properties.
are irreducible, of degree one, and all homotheties.
We obtain also the following
Corollary. the index
is still valid.
Classification of representations of a semidirect product
, is abelian. . This is the so-called method of “little groups” of Wigner and Mackey.
be a representative system of the orbit of degree one which is equal to its own character.
with the canonical projection Finally, we construct the tensor product
Thus, we achieve the following result:
Proposition.
is irreducible.
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 .
Representation ring
is defined as the abelian group
With the multiplication provided by the tensor product are called virtual representations.
The character defines a ring homomorphism with complex values
is injective are called virtual characters.
As the irreducible characters form an orthonormal basis induces an isomorphism
This isomorphism is defined on a basis out of elementary tensors and extended bilinearly.
and the virtual characters correspond to the virtual representations in an optimal manner.
According to the Frobenius reciprocity is an ideal
is an ideal
–linear maps:
up to isomorphism.
is the representation of the direct product as was shown in a previous section is the tensor product–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 is finite.
is finitely generated as a group, the first point can be rephrased as follows:
gives two proofs of this theorem. For example, since G is a linear combination with rational coefficients of characters induced by characters of cyclic subgroups 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.
is surjective. Brauer's induction theorem 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 and a p–group. In other words, every character 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. (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
please refer to .
is called real is called the complexification
be a finite, non-abelian subgroup of the group
can be identified with the units of the quaternions is not real-valued, but has a real-valued character:
Thus, the character of the representation is real.
Lemma. is real if and only if there exists a nondegenerate symmetric bilinear form
we still haven't classified all the irreducible real representations. But we achieve the following:
Definition. A quaternionic Thus, a skew-symmetric
Theorem. is one and only one of the following:
–invariant nondegenerate
–invariant nondegenerate symmetric bilinear form.
–invariant skew-symmetric nondegenerate bilinear form.
Representations of particular groups
Symmetric groups
Representation of the symmetric groups (and therefore, by the above, irreducible representations) correspond to partitionss of n 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.
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
, as n; the above-mentioned induction process gets replaced by so-called parabolic induction. Instead, new building blocks, known as cuspidal representations, are needed.
and more generally, representations of finite groups of Lie type have been thoroughly studied. . 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.
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 adeless, see .
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 section please consult and .
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 which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again.
is a continuous group homomorphism
into a Banach space into the set of all bijective bounded linear operators 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 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
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
is compact, it is possible to show that this measure is also right-translation-invariant, i.e. it also applies
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:
is unitary, the two representations are called unitary equivalent.
–invariant inner product is an inner product on a Hilbert space then
Thus, we can assume every representation on a Hilbert space to be unitary.
It is called left-regular representation. The right-regular representation are dual to each other.
is infinite, these representations have no finite degree. The left- and right-regular representation
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 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
is defined by the property
is again a continuous group homomorphism and thus a representation.
is irreducible.
By transferring the results of the section decompositions to compact groups, we obtain the following theorems:
Theorem. of a compact group into a Hilbert space is finite-dimensional and there exists an inner product 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.
to be the subspace
Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal.
Theorem.
passes through all the isomorphism classes of the irreducible representations.
is for compact groups given by
Projection formula
we define
–linear. Let
is defined as endomorphism by having the property
–linear, because of
where we used the invariance of the Haar measure.
Proposition.
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:
Then every bounded operator
Definition. The formula
Likewise
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. respectively. Then the following is valid
Theorem.
Irreducibility Criterion. is irreducible.
form an orthonormal set with respect to this inner product.
Corollary.–times in the left-regular representation.
Lemma. be a compact group. Then the following statements are equivalent:
is abelian.
Orthonormal Property. form an orthonormal basis with respect to this inner product.
orthogonal to all the irreducible characters.
However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information.
The induced representation
is a closed subgroup of finite index the definition of the induced representation for finite groups may be adopted.
is defined as follows:
The norm is given by
The induced representation is then again a unitary representation.
then there exists a canonical isomorphism
The Frobenius reciprocity 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. be an orthonormal basis We define the matrix coefficients Then we have the following orthonormal basis:
We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups:
The Peter-Weyl Theorem (Second version). A proof of this theorem and more information regarding the representation theory of compact groups may be found in –isomorphism
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
Notes
References