The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.
Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over
Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms.
Definition
Linear representations Let be a –vector space and a finite group. A linear representation of is a group homomorphism Here is notation for a general linear group, and for an automorphism group. This means that a linear representation is a map which satisfies for all The vector space is called a 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 section. As in most cases only a finite number of vectors in is of interest, it is sufficient to study the subrepresentation generated by these vectors. The representation space of this subrepresentation is then finite-dimensional.
The degree of a representation is the dimension of its representation space The notation is sometimes used to denote the degree of a representation
Examples The trivial representation is given by for all
A representation of degree of a group is a homomorphism into the multiplicative group As every element of is of finite order, the values of are roots of unity. For example, let be a nontrivial linear representation. Since is a group homomorphism, it has to satisfy Because generates is determined by its value on And as is nontrivial, Thus, we achieve the result that the image of under has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. In other words, has to be one of the following three maps:
Let and let be the group homomorphism defined by:
In this case is a linear representation of of degree
Permutation representation
Let be a finite set and let be a group acting on Denote by the group of all permutations on with the composition as group multiplication.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations—where groups act on vector spaces instead of on arbitrary finite sets—we have to proceed in a different way. In order to construct the permutation representation, we need a vector space with A basis of can be indexed by the elements of The permutation representation is the group homomorphism given by for all All linear maps are uniquely defined by this property.
Example. Let and Then acts on via The associated linear representation is with for
Left- and right-regular representation
Let be a group and be a vector space of dimension with a basis indexed by the elements of The left-regular representation is a special case of the permutation representation by choosing This means for all Thus, the family of images of are a basis of The degree of the left-regular representation is equal to the order of the group.
The right-regular representation is defined on the same vector space with a similar homomorphism: In the same way as before is a basis of Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order of
These two 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.
The involution
turns into a –algebra. We have
A representation of a group extends to a –algebra homomorphism by Since multiplicativity is a characteristic property of algebra homomorphisms, satisfies If is unitary, we also obtain For the definition of a unitary representation, please refer to the 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 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.Proof. Suppose is nonzero. Then is valid for all Therefore, we obtain for all and And we know now, that is –invariant. Since is irreducible and we conclude Now let This means, there exists such that and we have Thus, we deduce, that is a –invariant subspace. Because is nonzero and is irreducible, we have Therefore, is an isomorphism and the first statement is proven.
Suppose now that Since our base field is we know that has at least one eigenvalue Let then and we have for all According to the considerations above this is only possible, if i.e.
Properties Two representations are called equivalent or isomorphic, if there exists a –linear vector space isomorphism between the representation spaces. In other words, they are isomorphic if there exists a bijective linear map such that for all In particular, equivalent representations have the same degree.
A representation is called faithful when is injective. In this case induces an isomorphism between and the image As the latter is a subgroup of we can regard via as subgroup of
We can restrict the range as well as the domain:
Let be a subgroup of Let be a linear representation of We denote by the restriction of to the subgroup
If there is no danger of confusion, we might use only or in short
The notation or in short is also used to denote the restriction of the representation of onto
Let be a function on We write or shortly for the restriction to the subgroup
It can be proven that the number of irreducible representations of a group (or correspondingly the number of simple –modules) equals the number of conjugacy classes of
A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.
For the definition of the direct sum of representations please refer to the section on direct sums of representations.
A representation is called isotypic if it is a direct sum of pairwise isomorphic irreducible representations.
Let be a given representation of a group Let be an irreducible representation of The –isotype of is defined as the sum of all irreducible subrepresentations of isomorphic to
Every vector space over can be provided with an inner product. A representation of a group in a vector space endowed with an inner product is called unitary if is unitary for every This means that in particular every is diagonalizable. For more details see the article on unitary representations.
A representation is unitary with respect to a given inner product if and only if the inner product is invariant with regard to the induced operation of i.e. if and only if holds for all
A given inner product can be replaced by an invariant inner product by exchanging with