Representations of finite groups, particularly on vector spaces
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.
Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over
Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms.
Definition
Linear representations
Let
be a
–vector space and
a finite group. A linear representation of
is a group homomorphism
Here
is notation for a general linear group, and
for an automorphism group. This means that a linear representation is a map
which satisfies
for all
The vector space
is called representation space of
Often the term representation of
is also used for the representation space
The representation of a group in a module instead of a vector space is also called a linear representation.
We write
for the representation
of
Sometimes we use the notation
if it is clear to which representation the space
belongs.
In this article we will restrict ourselves to the study of finite-dimensional representation spaces, except for the last chapter. As in most cases only a finite number of vectors in
is of interest, it is sufficient to study the subrepresentation generated by these vectors. The representation space of this subrepresentation is then finite-dimensional.
The degree of a representation is the dimension of its representation space
The notation
is sometimes used to denote the degree of a representation
Examples
The trivial representation is given by
for all
A representation of degree
of a group
is a homomorphism into the multiplicative group
As every element of
is of finite order, the values of
are roots of unity. For example, let
be a nontrivial linear representation. Since
is a group homomorphism, it has to satisfy
Because
generates
is determined by its value on
And as
is nontrivial,
Thus, we achieve the result that the image of
under
has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. In other words,
has to be one of the following three maps:
![{\displaystyle {\begin{cases}\rho _{1}({0})=1\\\rho _{1}({1})=i\\\rho _{1}({2})=-1\\\rho _{1}({3})=-i\end{cases}}\qquad {\begin{cases}\rho _{2}({0})=1\\\rho _{2}({1})=-1\\\rho _{2}({2})=1\\\rho _{2}({3})=-1\end{cases}}\qquad {\begin{cases}\rho _{3}({0})=1\\\rho _{3}({1})=-i\\\rho _{3}({2})=-1\\\rho _{3}({3})=i\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26870c5a70e0129aec0cea03e0277f9d39cf6ae3)
Let
and let
be the group homomorphism defined by:
![{\displaystyle \rho ({0},{0})={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad \rho ({1},{0})={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}},\quad \rho ({0},{1})={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \rho ({1},{1})={\begin{pmatrix}0&-1\\-1&0\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f542ff73429b8fbf9292caa7114bd8bd7e2d19)
In this case
is a linear representation of
of degree
Permutation representation
Let
be a finite set and let
be a group acting on
Denote by
the group of all permutations on
with the composition as group multiplication.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way. In order to construct the permutation representation, we need a vector space
with
A basis of
can be indexed by the elements of
The permutation representation is the group homomorphism
given by
for all
All linear maps
are uniquely defined by this property.
Example. Let
and
Then
acts on
via
The associated linear representation is
with
for
Left- and right-regular representation
Let
be a group and
be a vector space of dimension
with a basis
indexed by the elements of
The left-regular representation is a special case of the permutation representation by choosing
This means
for all
Thus, the family
of images of
are a basis of
The degree of the left-regular representation is equal to the order of the group.
The right-regular representation is defined on the same vector space with a similar homomorphism:
In the same way as before
is a basis of
Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order of
Both representations are isomorphic via
For this reason they are not always set apart, and often referred to as "the" regular representation.
A closer look provides the following result: A given linear representation
is isomorphic to the left-regular representation if and only if there exists a
such that
is a basis of
Example. Let
and
with the basis
Then the left-regular representation
is defined by
for
The right-regular representation is defined analogously by
for
Representations, modules and the convolution algebra
Let
be a finite group, let
be a commutative ring and let
be the group algebra of
over
This algebra is free and a basis can be indexed by the elements of
Most often the basis is identified with
. Every element
can then be uniquely expressed as
with
.
The multiplication in
extends that in
distributively.
Now let
be a
–module and let
be a linear representation of
in
We define
for all
and
. By linear extension
is endowed with the structure of a left-
–module. Vice versa we obtain a linear representation of
starting from a
–module
. Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. Therefore, these terms may be used interchangeably.[1][2] This is an example of an isomorphism of categories.
Suppose
In this case the left
–module given by
itself corresponds to the left-regular representation. In the same way
as a right
–module corresponds to the right-regular representation.
In the following we will define the convolution algebra: Let
be a group, the set
is a
–vector space with the operations addition and scalar multiplication then this vector space is isomorphic to
The convolution of two elements
defined by
![{\displaystyle f*h(s):=\sum _{t\in G}f(t)h(t^{-1}s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c726af50e76f6604c6665bc7d82a470526d35f)
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
![{\displaystyle \delta _{s}(t)={\begin{cases}1&t=s\\0&{\text{otherwise.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/055f1c23f72eb5687e89056e09448779df69a050)
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
![{\displaystyle f^{*}(s)={\overline {f(s^{-1})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/433597e5b1c1dd0dc7522bc514e29c202bbb0d41)
turns
into a
–algebra. We have
A representation
of a group
extends to a
–algebra homomorphism
by
Since multiplicativity is a characteristic property of algebra homomorphisms,
satisfies
If
is unitary, we also obtain
For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary.
Using the convolution algebra we can implement a Fourier transformation on a group
In the area of harmonic analysis it is shown that the following definition is consistent with the definition of the Fourier transformation on
Let
be a representation and let
be a
-valued function on
. The Fourier transform
of
is defined as
![{\displaystyle {\hat {f}}(\rho )=\sum _{s\in G}f(s)\rho (s).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519a49a51a24228a3a2a5f492ab43b0662063947)
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
:
![](//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Equivariant_map.svg/200px-Equivariant_map.svg.png)
Such a map is also called
–linear, or an equivariant map. The kernel, the image and the cokernel of
are defined by default. The composition of equivariant maps is again an equivariant map. There is a category of representations with equivariant maps as its morphisms. They are again
–modules. Thus, they provide representations of
due to the correlation described in the previous section.
Irreducible representations and Schur's lemma
Let
be a linear representation of
Let
be a
-invariant subspace of
that is,
for all
and
. The restriction
is an isomorphism of
onto itself. Because
holds for all
this construction is a representation of
in
It is called subrepresentation of
Any representation V has at least two subrepresentations, namely the one consisting only of 0, and the one consisting of V itself. The representation is called an irreducible representation, if these two are the only subrepresentations. Some authors also call these representations simple, given that they are precisely the simple modules over the group algebra
.
Schur's lemma puts a strong constraint on maps between irreducible representations. If
and
are both irreducible, and
is a linear map such that
for all
, there is the following dichotomy:
- If
and
is a homothety (i.e.
for a
). More generally, if
and
are isomorphic, the space of G-linear maps is one-dimensional. - Otherwise, if the two representations are not isomorphic, F must be 0.[3]
Properties
Two representations
are called equivalent or isomorphic, if there exists a
–linear vector space isomorphism between the representation spaces. In other words, they are isomorphic if there exists a bijective linear map
such that
for all
In particular, equivalent representations have the same degree.
A representation
is called faithful when
is injective. In this case
induces an isomorphism between
and the image
As the latter is a subgroup of
we can regard
via
as subgroup of
We can restrict the range as well as the domain:
Let
be a subgroup of
Let
be a linear representation of
We denote by
the restriction of
to the subgroup
If there is no danger of confusion, we might use only
or in short
The notation
or in short
is also used to denote the restriction of the representation
of
onto
Let
be a function on
We write
or shortly
for the restriction to the subgroup
It can be proven that the number of irreducible representations of a group
(or correspondingly the number of simple
–modules) equals the number of conjugacy classes of
A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.
For the definition of the direct sum of representations please refer to the section on direct sums of representations.
A representation is called isotypic if it is a direct sum of pairwise isomorphic irreducible representations.
Let
be a given representation of a group
Let
be an irreducible representation of
The
–isotype
of
is defined as the sum of all irreducible subrepresentations of
isomorphic to
Every vector space over
can be provided with an inner product. A representation
of a group
in a vector space endowed with an inner product is called unitary if
is unitary for every
This means that in particular every
is diagonalizable. For more details see the article on unitary representations.
A representation is unitary with respect to a given inner product if and only if the inner product is invariant with regard to the induced operation of
i.e. if and only if
holds for all
A given inner product
can be replaced by an invariant inner product by exchanging
with
![{\displaystyle \sum _{t\in G}(\rho (t)v|\rho (t)u).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2fd6283ae02105c115f80fce8b29f684634726)
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:
![{\displaystyle \rho (\mu )=\left({\begin{array}{ccc}\cos({\frac {2\pi }{3}})&0&-\sin({\frac {2\pi }{3}})\\0&1&0\\\sin({\frac {2\pi }{3}})&0&\cos({\frac {2\pi }{3}})\end{array}}\right),\,\,\,\,\rho (\nu )=\left({\begin{array}{ccc}-1&0&0\\0&-1&0\\0&0&1\end{array}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350613369a4692031080abbfc17bbfde1f2284e3)
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
![{\displaystyle \nu \mapsto {\begin{pmatrix}-1&0\\0&1\end{pmatrix}},\,\,\,\,\mu \mapsto {\begin{pmatrix}\cos({\frac {2\pi }{3}})&-\sin({\frac {2\pi }{3}})\\\sin({\frac {2\pi }{3}})&\cos({\frac {2\pi }{3}})\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72de90f0a18639fffc21ef54bcd255e9ca5dc31e)
This subrepresentation is also irreducible. That means, the original representation is completely reducible:
![{\displaystyle \rho =\rho |_{\mathbb {C} e_{2}}\oplus \rho |_{\mathbb {C} e_{1}\oplus \mathbb {C} e_{3}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f459f74ad5378eb7add9482c9445e1a39a9c793)
Both subrepresentations are isotypic and are the two only non-zero isotypes of
The representation
is unitary with regard to the standard inner product on
because
and
are unitary.
Let
be any vector space isomorphism. Then
which is defined by the equation
for all
is a representation isomorphic to
By restricting the domain of the representation to a subgroup, e.g.
we obtain the representation
This representation is defined by the image
whose explicit form is shown above.
Constructions
The dual representation
Let
be a given representation. The dual representation or contragredient representation
is a representation of
in the dual vector space of
It is defined by the property
![{\displaystyle \forall s\in G,v\in V,\alpha \in V^{*}:\qquad \left(\rho ^{*}(s)\alpha \right)(v)=\alpha \left(\rho \left(s^{-1}\right)v\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/701fdec3a4942805e1eea1f62d6de9a7f579b2c8)
With regard to the natural pairing
between
and
the definition above provides the equation:
![{\displaystyle \forall s\in G,v\in V,\alpha \in V^{*}:\qquad \langle \rho ^{*}(s)(\alpha ),\rho (s)(v)\rangle =\langle \alpha ,v\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286a84d1535d5e6a8d7757c9b0651de1de1f03ad)
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
![{\displaystyle \forall s_{1}\in G_{1},s_{2}\in G_{2},v_{1}\in V_{1},v_{2}\in V_{2}:\qquad {\begin{cases}\rho _{1}\oplus \rho _{2}:G_{1}\times G_{2}\to {\text{GL}}(V_{1}\oplus V_{2})\\[4pt](\rho _{1}\oplus \rho _{2})(s_{1},s_{2})(v_{1},v_{2}):=\rho _{1}(s_{1})v_{1}\oplus \rho _{2}(s_{2})v_{2}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac4efc7d528b4d9d0c811273c4589cd1fa17962)
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):
![{\displaystyle {\begin{cases}\rho _{1}:\mathbb {Z} /2\mathbb {Z} \to {\text{GL}}_{2}(\mathbb {C} )\\[4pt]\rho _{1}(1)={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\end{cases}}\qquad \qquad {\begin{cases}\rho _{2}:\mathbb {Z} /3\mathbb {Z} \to {\text{GL}}_{3}(\mathbb {C} )\\[6pt]\rho _{2}(1)={\begin{pmatrix}1&0&\omega \\0&\omega &0\\0&0&\omega ^{2}\end{pmatrix}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3f54de71e01a5dc4f85b982e627c8916756549)
Then
![{\displaystyle {\begin{cases}\rho _{1}\oplus \rho _{2}:\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /3\mathbb {Z} \to {\text{GL}}\left(\mathbb {C} ^{2}\oplus \mathbb {C} ^{3}\right)\\[6pt]\left(\rho _{1}\oplus \rho _{2}\right)(k,l)={\begin{pmatrix}\rho _{1}(k)&0\\0&\rho _{2}(l)\end{pmatrix}}&k\in \mathbb {Z} /2\mathbb {Z} ,l\in \mathbb {Z} /3\mathbb {Z} \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14d1e108f72cc9bb9e0f65a57b7bc3baefc92d60)
As it is sufficient to consider the image of the generating element, we find that
![{\displaystyle (\rho _{1}\oplus \rho _{2})(1,1)={\begin{pmatrix}0&-i&0&0&0\\i&0&0&0&0\\0&0&1&0&\omega \\0&0&0&\omega &0\\0&0&0&0&\omega ^{2}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10eba594310a48e5193c9c1e901cb3ebb247ed7)
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:
![{\displaystyle {\begin{cases}\rho _{1}:\mathbb {Z} /2\mathbb {Z} \to {\text{GL}}_{2}(\mathbb {C} )\\[4pt]\rho _{1}(1)={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\end{cases}}\qquad \qquad {\begin{cases}\rho _{2}:\mathbb {Z} /3\mathbb {Z} \to {\text{GL}}_{3}(\mathbb {C} )\\[6pt]\rho _{2}(1)={\begin{pmatrix}1&0&\omega \\0&\omega &0\\0&0&\omega ^{2}\end{pmatrix}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3f54de71e01a5dc4f85b982e627c8916756549)
The outer tensor product
![{\displaystyle {\begin{cases}\rho _{1}\otimes \rho _{2}:\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /3\mathbb {Z} \to {\text{GL}}(\mathbb {C} ^{2}\otimes \mathbb {C} ^{3})\\(\rho _{1}\otimes \rho _{2})(k,l)=\rho _{1}(k)\otimes \rho _{2}(l)&k\in \mathbb {Z} /2\mathbb {Z} ,l\in \mathbb {Z} /3\mathbb {Z} \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d49b93b200a889dc837e3d0958f92070f48542a)
Using the standard basis of
we have the following for the generating element:
![{\displaystyle \rho _{1}\otimes \rho _{2}(1,1)=\rho _{1}(1)\otimes \rho _{2}(1)={\begin{pmatrix}0&0&0&-i&0&-i\omega \\0&0&0&0&-i\omega &0\\0&0&0&0&0&-i\omega ^{2}\\i&0&i\omega &0&0&0\\0&i\omega &0&0&0&0\\0&0&i\omega ^{2}&0&0&0\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6e48acfbd93b3ebb30e7dc492afd555efaf55)
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 ![{\displaystyle s\in G,v\in V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aea3d0ee47a0f3ae90266e05e5e86c345707145)
- 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
![{\displaystyle {\text{Sym}}^{2}(V)=\{z\in V\otimes V:\vartheta (z)=z\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b9a6acacd73cf469f0b0826a6e1076ab5e1051)
![{\displaystyle {\text{Alt}}^{2}(V)=\bigwedge ^{2}V=\{z\in V\otimes V:\vartheta (z)=-z\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79fe730853d89bcd69c4045bf3a03fe140867362)
These subspaces are
–invariant and by this define subrepresentations which are called the symmetric square and the alternating square, respectively. These subrepresentations are also defined in
although in this case they are denoted wedge product
and symmetric product
In case that
the vector space
is in general not equal to the direct sum of these two products.
Decompositions
In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desirable. This can be achieved for finite groups as we will see in the following results. More detailed explanations and proofs may be found in [1] and [2].
- Theorem. (Maschke) Let
be a linear representation where
is a vector space over a field of characteristic zero. Let
be a
-invariant subspace of
Then the complement
of
exists in
and is
-invariant.
A subrepresentation and its complement determine a representation uniquely.
The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact – and therefore also of finite – groups:
- Theorem. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations.
Or in the language of
-modules: If
the group algebra
is semisimple, i.e. it is the direct sum of simple algebras.
Note that this decomposition is not unique. However, the number of how many times a subrepresentation isomorphic to a given irreducible representation is occurring in this decomposition is independent of the choice of decomposition.
The canonical decomposition
To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition is uniquely determined. It is called the canonical decomposition.
Let
be the set of all irreducible representations of a group
up to isomorphism. Let
be a representation of
and let
be the set of all isotypes of
The projection
corresponding to the canonical decomposition is given by
![{\displaystyle p_{j}={\frac {n_{j}}{g}}\sum _{t\in G}{\overline {\chi _{\tau _{j}}(t)}}\rho (t),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1c85bb24ed9d887e7dd24e339df5a012948382)
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
![{\displaystyle V^{G}:=\{v\in V:\rho (s)v=v\,\,\,\,\forall \,s\in G\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82099172dbda14f580b316acde45be99e9a1d67)
In general,
is not
-linear. We define
![{\displaystyle P:={\frac {1}{|G|}}\sum _{s\in G}\rho (s)\in {\text{End}}(V).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72057085a4d23df3764446204ab52b21313afbc5)
Then
is a
-linear map, because
![{\displaystyle \forall t\in G:\qquad \sum _{s\in G}\rho (s)=\sum _{s\in G}\rho (tst^{-1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e55c434048e0acacf3aba489d62f60b5f3c380)
- Proposition. The map
is a projection from
to ![{\displaystyle V^{G}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd165da373724cb9b98f27aff7829f10067e938)
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
![{\displaystyle \dim(V(1))=\dim(V^{G})=Tr(P)={\frac {1}{|G|}}\sum _{s\in G}\chi _{V}(s),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e8833abd6679a27954dffe6f488a9b43f0429d)
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
![{\displaystyle P={\frac {1}{|G|}}\sum _{s\in G}\pi (s)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9be803383b4f4e4fa73e29773f99326b889a6ff)
Let
be an orthonormal basis of
Then we have:
![{\displaystyle \sum _{s\in G}{\text{Tr}}(\pi (s))=\sum _{s\in G}\sum _{j=1}^{n}\langle \pi (s)e_{j},e_{j}\rangle =\sum _{j=1}^{n}\left\langle \sum _{s\in G}\pi (s)e_{j},e_{j}\right\rangle =0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba347d318dc52a2113bebfc309a3a385924c359)
Therefore, the following is valid for a nontrivial irreducible representation
:
![{\displaystyle \sum _{s\in G}\chi _{V}(s)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccd31703f6650d660686ff94c27299f87100b66)
Example. Let
be the permutation groups in three elements. Let
be a linear representation of
defined on the generating elements as follows:
![{\displaystyle \rho (1,2)={\begin{pmatrix}-1&2&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\end{pmatrix}},\quad \rho (1,3)={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}&0&0&0\\{\frac {1}{2}}&-1&0&0&0\\0&0&0&0&1\\0&0&0&1&0\\0&0&1&0&0\end{pmatrix}},\quad \rho (2,3)={\begin{pmatrix}0&-2&0&0&0\\-{\frac {1}{2}}&0&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f288b189469b5ba95ca490d963b8b90b98e3481)
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
![{\displaystyle \eta (1,2)={\begin{pmatrix}-1&2\\0&1\end{pmatrix}},\quad \eta (1,3)={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&-1\end{pmatrix}},\quad \eta (2,3)={\begin{pmatrix}0&-2\\-{\frac {1}{2}}&0\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca141a3745aa99a303d1a9babe1b6d946d777641)
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
![{\displaystyle \tau (1,2)={\begin{pmatrix}-1&0\\0&1\end{pmatrix}},\quad \tau (1,3)={\begin{pmatrix}{\frac {1}{2}}&{\frac {3}{2}}\\{\frac {1}{2}}&-{\frac {1}{2}}\end{pmatrix}},\quad \tau (2,3)={\begin{pmatrix}{\frac {1}{2}}&-{\frac {3}{2}}\\-{\frac {1}{2}}&-{\frac {1}{2}}\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/388156101f7557b8993ef7c0aac5887c8de7baf7)
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
![{\displaystyle M_{B}={\begin{pmatrix}2&2\\0&2\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2289eb659a3b42adfd10646ef6e3189f51e33f28)
A decomposition of
in irreducible subrepresentations is:
where
denotes the trivial representation and
![{\displaystyle \mathbb {C} ^{5}=\mathbb {C} (e_{1},e_{2})\oplus \mathbb {C} (e_{3}-e_{4},e_{3}+e_{4}-2e_{5})\oplus \mathbb {C} (e_{3}+e_{4}+e_{5})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1f9bfe42818cf9fd9c0155fd0d745f9fc85d75)
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
![{\displaystyle \rho =\rho _{1}\oplus 1,\qquad \mathbb {C} ^{5}=\mathbb {C} (e_{1},e_{2},e_{3}-e_{4},e_{3}+e_{4}-2e_{5})\oplus \mathbb {C} (e_{3}+e_{4}+e_{5}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2c938330e68f5242b8e4daea391bfc378ebc84)
The theorems above are in general not valid for infinite groups. This will be demonstrated by the following example: let
![{\displaystyle G=\{A\in {\text{GL}}_{2}(\mathbb {C} )|\,A\,\,{\text{ is an upper triangular matrix}}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4661ff732e3aa335b45bca755c6f4690ffccfc7)
Together with the matrix multiplication
is an infinite group.
acts on
by matrix-vector multiplication. We consider the representation
for all
The subspace
is a
-invariant subspace. However, there exists no
-invariant complement to this subspace. The assumption that such a complement exists would entail that every matrix is diagonalizable over
This is known to be wrong and thus yields a contradiction.
The moral of the story is that if we consider infinite groups, it is possible that a representation - even one that is not irreducible - can not be decomposed into a direct sum of irreducible subrepresentations.
Character theory
Definitions
The character of a representation
is defined as the map
in which
denotes the trace of the linear map
[4]
Even though the character is a map between two groups, it is not in general a group homomorphism, as the following example shows.
Let
be the representation defined by:
![{\displaystyle \rho (0,0)={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad \rho (1,0)={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}},\quad \rho (0,1)={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \rho (1,1)={\begin{pmatrix}0&-1\\-1&0\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55878d31c0e59b79e0700e0874d3cf7bda5c36a8)
The character
is given by
![{\displaystyle \chi _{\rho }(0,0)=2,\quad \chi _{\rho }(1,0)=-2,\quad \chi _{\rho }(0,1)=\chi _{\rho }(1,1)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37313a963d844af3cb5cfea85ac92652bff09529)
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
![{\displaystyle \chi _{V}(s)=|\{x\in X|s\cdot x=x\}|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a48c660e039289aae69479e9e4fb49566742f9e)
For example,[5] the character of the regular representation
is given by
![{\displaystyle \chi _{R}(s)={\begin{cases}0&s\neq e\\|G|&s=e\end{cases}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/956b59a71c23cfb698ebffd9ea0ea4d3a78f4312)
where
denotes the neutral element of
Properties
A crucial property of characters is the formula
![{\displaystyle \chi (tst^{-1})=\chi (s),\,\,\forall \,s,t\in G.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6287c993dcb0feb7c7f76706f2ab58c3ef28a09)
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.
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.
![{\displaystyle \forall s,t\in G:\quad \varphi \left(sts^{-1}\right)=\varphi (t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f05f96c218e9534a078e394cc84b0e312b2301c0)
Note that every character is a class function, as the trace of a matrix is preserved under conjugation.
The set of all class functions is a
–algebra and is denoted by
. Its dimension is equal to the number of conjugacy classes of
Proofs of the following results of this chapter may be found in [1], [2] and [3].
An inner product can be defined on the set of all class functions on a finite group:
![{\displaystyle (f|h)_{G}={\frac {1}{|G|}}\sum _{t\in G}f(t){\overline {h(t)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466448782880bd0266de19c84797efad759a2d1e)
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.
![{\displaystyle (\chi _{i}|\chi _{j})={\begin{cases}1{\text{ if }}i=j\\0{\text{ otherwise }}\end{cases}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a350d5ec20286d27cf14500420ffa7188fd285ba)
- 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:
![{\displaystyle (f|h)_{G}={\frac {1}{|G|}}\sum _{t\in G}f(t){\overline {h(t)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466448782880bd0266de19c84797efad759a2d1e)
A symmetric bilinear form can also be defined on
![{\displaystyle \langle f,h\rangle _{G}={\frac {1}{|G|}}\sum _{t\in G}f(t)h(t^{-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5261a2bd1bf77829483058ceb75c05a8d2147442)
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: ![{\displaystyle (\xi |\chi )={\frac {\dim(V(\tau ))}{\dim(\tau )}}=\langle V,W\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/252600a428fb658b28badfe3847eec79cdf9f908)
- and thus
![{\displaystyle \dim(V(\tau ))=\dim(\tau )(\xi |\chi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7da6f1cd2f919b716bb1d06eecfc0e9ad34346e2)
- 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:
![{\displaystyle |G|=\chi _{R}(e)=\dim(R)=\sum _{j}\dim \left((W_{j})^{\oplus (\chi _{W_{j}}|\chi _{R})}\right)=\sum _{j}(\chi _{W_{j}}|\chi _{R})\cdot \dim(W_{j})=\sum _{j}\dim(W_{j})^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db2109cd5ae436bbc169dc1529aade8fa73b0fc7)
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:
![{\displaystyle 0=\chi _{R}(s)=\sum _{j}\dim(W_{j})\cdot \chi _{W_{j}}(s).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0560ff2970a508247efe823923040bc6b57ebc30)
Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations:
The Fourier inversion formula:
![{\displaystyle f(s)={\frac {1}{|G|}}\sum _{\rho {\text{ irr. rep. of }}G}\dim(V_{\rho })\cdot {\text{Tr}}(\rho (s^{-1})\cdot {\hat {f}}(\rho )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05b5c66631131cfb2d283c1e2966ae7929ffaf97)
In addition, the Plancherel formula holds:
![{\displaystyle \sum _{s\in G}f(s^{-1})h(s)={\frac {1}{|G|}}\sum _{\rho \,\,{\text{ irred.}}{\text{ rep.}}{\text{ of }}G}\dim(V_{\rho })\cdot {\text{Tr}}({\hat {f}}(\rho ){\hat {h}}(\rho )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ff5ac9bdb1b8de85ab5404903aa49ba4fb49f6)
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 ![{\displaystyle 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24)
The induced representation
As was shown in the section on properties of linear representations, we can - by restriction - obtain a representation of a subgroup starting from a representation of a group. Naturally we are interested in the reverse process: Is it possible to obtain the representation of a group starting from a representation of a subgroup? We will see that the induced representation defined below provides us with the necessary concept. Admittedly, this construction is not inverse but rather adjoint to the restriction.
Definitions
Let
be a linear representation of
Let
be a subgroup and
the restriction. Let
be a subrepresentation of
We write
to denote this representation. Let
The vector space
depends only on the left coset
of
Let
be a representative system of
then
![{\displaystyle \sum _{r\in R}\rho (r)(W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c666d658a3ff0f60c288c9156b4367be93ab2f)
is a subrepresentation of
A representation
of
in
is called induced by the representation
of
in
if
![{\displaystyle V_{\rho }=\bigoplus _{r\in R}W_{r}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a52cc62cd0a2ecaead9cb727f27cc866037c2db7)
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
![{\displaystyle \sum _{r\in R}w_{r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d689f77d840614016832ccb41b4e4cfe0d4f38)
where
for every
We denote the representation
of
which is induced by the representation
of
as
or in short
if there is no danger of confusion. The representation space itself is frequently used instead of the representation map, i.e.
or
if the representation
is induced by
Alternative description of the induced representation
By using the group algebra we obtain an alternative description of the induced representation:
Let
be a group,
a
–module and
a
–submodule of
corresponding to the subgroup
of
We say that
is induced by
if
in which
acts on the first factor:
for all
Properties
The results introduced in this section will be presented without proof. These may be found in [1] and [2].
- Uniqueness and existence of the induced representation. Let
be a linear representation of a subgroup
of
Then there exists a linear representation
of
which is induced by
Note that this representation is unique up to isomorphism.
- Transitivity of induction. Let
be a representation of
and let
be an ascending series of groups. Then we have ![{\displaystyle {\text{Ind}}_{G}^{K}({\text{Ind}}_{H}^{G}(W))\cong {\text{Ind}}_{H}^{K}(W).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7fb30868f7aed52a9466ce09841e6030047c1a)
- 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 ![{\displaystyle s\in G.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/083d6db517ca550e316039b3232a08c27c5198ef)
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
![{\displaystyle \varphi '(s)={\frac {1}{|H|}}\sum _{t\in G \atop t^{-1}st\in H}^{}\varphi (t^{-1}st).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e08b5e8aaa55c62c5b9b136457f363911d3064)
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 ![{\displaystyle G.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a)
- Lemma. If
is a class function on
and
is a class function on
then we have: ![{\displaystyle {\text{Ind}}(\psi \cdot {\text{Res}}\varphi )=({\text{Ind}}\psi )\cdot \varphi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e507f284e55a265a4a9900f30401adbd3713eca4)
- 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 ![{\displaystyle \forall t\in G:\qquad \chi _{\rho }(t)=\sum _{r\in R, \atop r^{-1}tr\in H}^{}\chi _{\theta }(r^{-1}tr)={\frac {1}{|H|}}\sum _{s\in G, \atop s^{-1}ts\in H}^{}\chi _{\theta }(s^{-1}ts).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a122d5de92c6ce67a1dd9e6f528d36ba2de171de)
Frobenius reciprocity
As a preemptive summary, the lesson to take from Frobenius reciprocity is that the maps
and
are adjoint to each other.
Let
be an irreducible representation of
and let
be an irreducible representation of
then the Frobenius reciprocity tells us that
is contained in
as often as
is contained in
- Frobenius reciprocity. If
and
we have ![{\displaystyle \langle \psi ,{\text{Res}}(\varphi )\rangle _{H}=\langle {\text{Ind}}(\psi ),\varphi \rangle _{G}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fca979a695ba41db7ce95ef2da2eb92fb972f2bf)
This statement is also valid for the inner product.
Mackey's irreducibility criterion
George Mackey established a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation.
Two representations
and
of a group
are called disjoint, if they have no irreducible component in common, i.e. if
Let
be a group and let
be a subgroup. We define
for
Let
be a representation of the subgroup
This defines by restriction a representation
of
We write
for
We also define another representation
of
by
These two representations are not to be confused.
- Mackey's irreducibility criterion. The induced representation
is irreducible if and only if the following conditions are satisfied:
is irreducible - For each
the two representations
and
of
are disjoint.[6]
For the case of
normal, we have
and
. Thus we obtain the following:
- Corollary. Let
be a normal subgroup of
Then
is irreducible if and only if
is irreducible and not isomorphic to the conjugates
for ![{\displaystyle s\notin H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c98b9b0963bd8f415adfde372a15a33b51234d5)
Applications to special groups
In this section we present some applications of the so far presented theory to normal subgroups and to a special group, the semidirect product of a subgroup with an abelian normal subgroup.
- Proposition. Let
be a normal subgroup of the group
and let
be an irreducible representation of
Then one of the following statements has to be valid: - either there exists a proper subgroup
of
containing
, and an irreducible representation
of
which induces
, - or
is an isotypic
-module.
- Proof. Consider
as a
-module, and decompose it into isotypes as
. If this decomposition is trivial, we are in the second case. Otherwise, the larger
-action permutes these isotypic modules; because
is irreducible as a
-module, the permutation action is transitive (in fact primitive). Fix any
; the stabilizer in
of
is elementarily seen to exhibit the claimed properties. ![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Note that if
is abelian, then the isotypic modules of
are irreducible, of degree one, and all homotheties.
We obtain also the following
- Corollary. Let
be an abelian normal subgroup of
and let
be any irreducible representation of
We denote with
the index of
in
Then
[1]
If
is an abelian subgroup of
(not necessarily normal), generally
is not satisfied, but nevertheless
is still valid.
Classification of representations of a semidirect product
In the following, let
be a semidirect product such that the normal semidirect factor,
, is abelian. The irreducible representations of such a group
can be classified by showing that all irreducible representations of
can be constructed from certain subgroups of
. This is the so-called method of “little groups” of Wigner and Mackey.
Since
is abelian, the irreducible characters of
have degree one and form the group
The group
acts on
by
for
Let
be a representative system of the orbit of
in
For every
let
This is a subgroup of
Let
be the corresponding subgroup of
We now extend the function
onto
by
for
Thus,
is a class function on
Moreover, since
for all
it can be shown that
is a group homomorphism from
to
Therefore, we have a representation of
of degree one which is equal to its own character.
Let now
be an irreducible representation of
Then we obtain an irreducible representation
of
by combining
with the canonical projection
Finally, we construct the tensor product of
and
Thus, we obtain an irreducible representation
of
To finally obtain the classification of the irreducible representations of
we use the representation
of
which is induced by the tensor product
Thus, we achieve the following result:
- Proposition.
is irreducible. - If
and
are isomorphic, then
and additionally
is isomorphic to ![{\displaystyle \rho '.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa47785f4d4a551d64efafb7e19621d4d9a220f)
- Every irreducible representation of
is isomorphic to one of the ![{\displaystyle \theta _{j,\rho }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36198b5a78f84a2bbd3f2f20904ce3926cb2d893)
Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in [1].
In other words, we classified all irreducible representations of
Representation ring
The representation ring of
is defined as the abelian group
![{\displaystyle R(G)=\left\{\left.\sum _{j=1}^{m}a_{j}\tau _{j}\right|\tau _{1},\ldots ,\tau _{m}{\text{ all irreducible representations of }}G{\text{ up to isomorphism}},a_{j}\in \mathbb {Z} \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9068b1b50b09d4aa91e3723c1795502b78662f)
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
![{\displaystyle {\begin{cases}\chi :R(G)\to \mathbb {C} _{\text{class}}(G)\\\sum a_{j}\tau _{j}\mapsto \sum a_{j}\chi _{j}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3115b3a61fa807b2ef810d01008776c820cf320d)
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
![{\displaystyle \chi _{\mathbb {C} }:R(G)\otimes \mathbb {C} \to \mathbb {C} _{\text{class}}(G).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d25a4fd9f46e2a5eedf0bdcc1904490536a0040f)
This isomorphism is defined on a basis out of elementary tensors
by
respectively
and extended bilinearly.
We write
for the set of all characters of
and
to denote the group generated by
i.e. the set of all differences of two characters. It then holds that
and
Thus, we have
and the virtual characters correspond to the virtual representations in an optimal manner.
Since
holds,
is the set of all virtual characters. As the product of two characters provides another character,
is a subring of the ring
of all class functions on
Because the
form a basis of
we obtain, just as in the case of
an isomorphism
Let
be a subgroup of
The restriction thus defines a ring homomorphism
which will be denoted by
or
Likewise, the induction on class functions defines a homomorphism of abelian groups
which will be written as
or in short
According to the Frobenius reciprocity, these two homomorphisms are adjoint with respect to the bilinear forms
and
Furthermore, the formula
shows that the image of
is an ideal of the ring
By the restriction of representations, the map
can be defined analogously for
and by the induction we obtain the map
for
Due to the Frobenius reciprocity, we get the result that these maps are adjoint to each other and that the image
is an ideal of the ring
If
is a commutative ring, the homomorphisms
and
may be extended to
–linear maps:
![{\displaystyle {\begin{cases}A\otimes {\text{Res}}:A\otimes R(G)\to A\otimes R(H)\\\left(a\otimes \sum a_{j}\tau _{j}\right)\mapsto \left(a\otimes \sum a_{j}{\text{Res}}(\tau _{j})\right)\end{cases}},\qquad {\begin{cases}A\otimes {\text{Ind}}:A\otimes R(H)\to A\otimes R(G)\\\left(a\otimes \sum a_{j}\eta _{j}\right)\mapsto \left(a\otimes \sum a_{j}{\text{Ind}}(\eta _{j})\right)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c76b3ebf35186174d54e8bc4f03cfc606fd94fa)
in which
are all the irreducible representations of
up to isomorphism.
With
we obtain in particular that
and
supply homomorphisms between
and
Let
and
be two groups with respective representations
and
Then,
is the representation of the direct product
as was shown in a previous section. Another result of that section was that all irreducible representations of
are exactly the representations
where
and
are irreducible representations of
and
respectively. This passes over to the representation ring as the identity
in which
is the tensor product of the representation rings as
–modules.
Induction theorems
Induction theorems relate the representation ring of a given finite group G to representation rings of a family X consisting of some subsets H of G. More precisely, for such a collection of subgroups, the induction functor yields a map
; induction theorems give criteria for the surjectivity of this map or closely related ones.
Artin's induction theorem is the most elementary theorem in this group of results. It asserts that the following are equivalent:
- The cokernel of
is finite.
is the union of the conjugates of the subgroups belonging to
i.e. ![{\displaystyle G=\bigcup _{H\in X \atop s\in G}sHs^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e98fcf49ac097201ed0fa715404e046e89e90d96)
Since
is finitely generated as a group, the first point can be rephrased as follows:
- For each character
of
there exist virtual characters
and an integer
such that ![{\displaystyle d\cdot \chi =\sum _{H\in X}{\text{Ind}}_{H}^{G}(\chi _{H}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4d1bc7a29c7868ce8e18289254c6fc9fc2bc2f)
Serre (1977) gives two proofs of this theorem. For example, since G is the union of its cyclic subgroups, every character of
is a linear combination with rational coefficients of characters induced by characters of cyclic subgroups of
Since the representations of cyclic groups are well-understood, in particular the irreducible representations are one-dimensional, this gives a certain control over representations of G.
Under the above circumstances, it is not in general true that
is surjective. Brauer's induction theorem asserts that
is surjective, provided that X is the family of all elementary subgroups. Here a group H is elementary if there is some prime p such that H is the direct product of a cyclic group of order prime to
and a
–group. In other words, every character of
is a linear combination with integer coefficients of characters induced by characters of elementary subgroups. The elementary subgroups H arising in Brauer's theorem have a richer representation theory than cyclic groups, they at least have the property that any irreducible representation for such H is induced by a one-dimensional representation of a (necessarily also elementary) subgroup
. (This latter property can be shown to hold for any supersolvable group, which includes nilpotent groups and, in particular, elementary groups.) This ability to induce representations from degree 1 representations has some further consequences in the representation theory of finite groups.
Real representations
For proofs and more information about representations over general subfields of
please refer to [2].
If a group
acts on a real vector space
the corresponding representation on the complex vector space
is called real (
is called the complexification of
). The corresponding representation mentioned above is given by
for all
Let
be a real representation. The linear map
is
-valued for all
Thus, we can conclude that the character of a real representation is always real-valued. But not every representation with a real-valued character is real. To make this clear, let
be a finite, non-abelian subgroup of the group
![{\displaystyle {\text{SU}}(2)=\left\{{\begin{pmatrix}a&b\\-{\overline {b}}&{\overline {a}}\end{pmatrix}}\ :\ |a|^{2}+|b|^{2}=1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2723698b7fb879714fee1973a8945907e8e1be6b)
Then
acts on
Since the trace of any matrix in
is real, the character of the representation is real-valued. Suppose
is a real representation, then
would consist only of real-valued matrices. Thus,
However the circle group is abelian but
was chosen to be a non-abelian group. Now we only need to prove the existence of a non-abelian, finite subgroup of
To find such a group, observe that
can be identified with the units of the quaternions. Now let
The following two-dimensional representation of
is not real-valued, but has a real-valued character:
![{\displaystyle {\begin{cases}\rho :G\to {\text{GL}}_{2}(\mathbb {C} )\\[4pt]\rho (\pm 1)={\begin{pmatrix}\pm 1&0\\0&\pm 1\end{pmatrix}},\quad \rho (\pm i)={\begin{pmatrix}\pm i&0\\0&\mp i\end{pmatrix}},\quad \rho (\pm j)={\begin{pmatrix}0&\pm i\\\pm i&0\end{pmatrix}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1951a54fb63c603e82bdbf5996648da4862bec5e)
Then the image of
is not real-valued, but nevertheless it is a subset of
Thus, the character of the representation is real.
- Lemma. An irreducible representation
of
is real if and only if there exists a nondegenerate symmetric bilinear form
on
preserved by ![{\displaystyle G.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a)
An irreducible representation of
on a real vector space can become reducible when extending the field to
For example, the following real representation of the cyclic group is reducible when considered over
![{\displaystyle {\begin{cases}\rho :\mathbb {Z} /m\mathbb {Z} \to {\text{GL}}_{2}(\mathbb {R} )\\[4pt]\rho (k)={\begin{pmatrix}\cos \left({\frac {2\pi ik}{m}}\right)&\sin \left({\frac {2\pi ik}{m}}\right)\\-\sin \left({\frac {2\pi ik}{m}}\right)&\cos \left({\frac {2\pi ik}{m}}\right)\end{pmatrix}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faa9aaf2bc6c2828a4f69cabe8b0ae5d3f1209b7)
Therefore, by classifying all the irreducible representations that are real over
we still haven't classified all the irreducible real representations. But we achieve the following:
Let
be a real vector space. Let
act irreducibly on
and let
If
is not irreducible, there are exactly two irreducible factors which are complex conjugate representations of
Definition. A quaternionic representation is a (complex) representation
which possesses a
–invariant anti-linear homomorphism
satisfying
Thus, a skew-symmetric, nondegenerate
–invariant bilinear form defines a quaternionic structure on
- Theorem. An irreducible representation
is one and only one of the following: - (i) complex:
is not real-valued and there exists no
–invariant nondegenerate bilinear form on ![{\displaystyle V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978)
- (ii) real:
a real representation;
has a
–invariant nondegenerate symmetric bilinear form. - (iii) quaternionic:
is real, but
is not real;
has a
–invariant skew-symmetric nondegenerate bilinear form.
Representations of particular groups
Symmetric groups
Representation of the symmetric groups
have been intensely studied. Conjugacy classes in
(and therefore, by the above, irreducible representations) correspond to partitions of n. For example,
has three irreducible representations, corresponding to the partitions
- 3; 2+1; 1+1+1
of 3. For such a partition, a Young tableau is a graphical device depicting a partition. The irreducible representation corresponding to such a partition (or Young tableau) is called a Specht module.
Representations of different symmetric groups are related: any representation of
yields a representation of
by induction, and vice versa by restriction. The direct sum of all these representation rings
![{\displaystyle \bigoplus _{n\geq 0}R(S_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acab7aafe5550072b94a7aecd624d955d7d52319)
inherits from these constructions the structure of a Hopf algebra which, it turns out, is closely related to symmetric functions.
Finite groups of Lie type
To a certain extent, the representations of the
, as n varies, have a similar flavor as for the
; the above-mentioned induction process gets replaced by so-called parabolic induction. However, unlike for
, where all representations can be obtained by induction of trivial representations, this is not true for
. Instead, new building blocks, known as cuspidal representations, are needed.
Representations of
and more generally, representations of finite groups of Lie type have been thoroughly studied. Bonnafé (2010) describes the representations of
. A geometric description of irreducible representations of such groups, including the above-mentioned cuspidal representations, is obtained by Deligne-Lusztig theory, which constructs such representation in the l-adic cohomology of Deligne-Lusztig varieties.
The similarity of the representation theory of
and
goes beyond finite groups. The philosophy of cusp forms highlights the kinship of representation theoretic aspects of these types of groups with general linear groups of local fields such as Qp and of the ring of adeles, see Bump (2004).
Outlook—Representations of compact groups
The theory of representations of compact groups may be, to some degree, extended to locally compact groups. The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms. For proofs, further information and for a more detailed insight which is beyond the scope of this chapter please consult [4] and [5].
Definition and properties
A topological group is a group together with a topology with respect to which the group composition and the inversion are continuous. Such a group is called compact, if any cover of
which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again.
Let
be a compact group and let
be a finite-dimensional
–vector space. A linear representation of
to
is a continuous group homomorphism
i.e.
is a continuous function in the two variables
and
A linear representation of
into a Banach space
is defined to be a continuous group homomorphism of
into the set of all bijective bounded linear operators on
with a continuous inverse. Since
we can do without the last requirement. In the following, we will consider in particular representations of compact groups in Hilbert spaces.
Just as with finite groups, we can define the group algebra and the convolution algebra. However, the group algebra provides no helpful information in the case of infinite groups, because the continuity condition gets lost during the construction. Instead the convolution algebra
takes its place.
Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart to the summation over a finite group:
Existence and uniqueness of the Haar measure
On a compact group
there exists exactly one measure
such that:
- It is a left-translation-invariant measure
![{\displaystyle \forall s\in G:\quad \int _{G}f(t)dt=\int _{G}f(st)dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5fcf439b6be094ceb353734b76130618069315)
- The whole group has unit measure:
![{\displaystyle \int _{G}dt=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7754187c9c6f7b37e01dfa644ac89860a3c0c7)
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
![{\displaystyle \forall s\in G:\quad \int _{G}f(t)dt=\int _{G}f(ts)dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0b2a6cd8d58b285b618d9591dc9d8644290d1e)
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
![{\displaystyle (v|u)_{\rho }=\int _{G}(\rho (t)v|\rho (t)u)dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27cc967d9bdbd4aa55eb8ca16b3de8bde6695432)
is a
–invariant inner product on
due to the properties of the Haar measure
Thus, we can assume every representation on a Hilbert space to be unitary.
Let
be a compact group and let
Let
be the Hilbert space of the square integrable functions on
We define the operator
on this space by
where
The map
is a unitary representation of
It is called left-regular representation. The right-regular representation is defined similarly. As the Haar measure of
is also right-translation-invariant, the operator
on
is given by
The right-regular representation is then the unitary representation given by
The two representations
and
are dual to each other.
If
is infinite, these representations have no finite degree. The left- and right-regular representation as defined at the beginning are isomorphic to the left- and right-regular representation as defined above, if the group
is finite. This is due to the fact that in this case
Constructions and decompositions
The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. The direct sum and the tensor product with a finite number of summands/factors are defined in exactly the same way as for finite groups. This is also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem saying that the irreducible representations of the product of two groups are (up to isomorphism) exactly the tensor product of the irreducible representations of the factor groups. First, we note that the direct product
of two compact groups is again a compact group when provided with the product topology. The Haar measure on the direct product is then given by the product of the Haar measures on the factor groups.
For the dual representation on compact groups we require the topological dual
of the vector space
This is the vector space of all continuous linear functionals from the vector space
into the base field. Let
be a representation of a compact group
in
The dual representation
is defined by the property
![{\displaystyle \forall v\in V,\forall v'\in V',\forall s\in G:\qquad \left\langle \pi '(s)v',\pi (s)v\right\rangle =\langle v',v\rangle :=v'(v).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d72b196d85a4e3e556f537e00423af5c1f2c767)
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
![{\displaystyle V_{\rho }(\tau )=\sum _{V_{\tau }\cong U\subset V_{\rho }}U.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01521a92ec8f1f9dff1672429f8b9e11d0e4ab05)
This is the sum of all invariant closed subspaces
which are
–isomorphic to
Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal.
- Theorem.
- (i)
is a closed invariant subspace of ![{\displaystyle V_{\rho }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d054e2cc2f4323c968c742ef0e3e2972c561f10b)
- (ii)
is
–isomorphic to the direct sum of copies of ![{\displaystyle V_{\tau }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4596237b72ba8c0bc6ff3b6ab52fc08c3b453815)
- (iii) Canonical decomposition:
is the direct Hilbert sum of the isotypes
in which
passes through all the isomorphism classes of the irreducible representations.
The corresponding projection to the canonical decomposition
in which
is an isotype of
is for compact groups given by
![{\displaystyle p_{\tau }(v)=n_{\tau }\int _{G}{\overline {\chi _{\tau }(t)}}\rho (t)(v)dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e663f3ffc93e26ac469b66abe88f9939fe3b94df)
where
and
is the character corresponding to the irreducible representation
For every representation
of a compact group
we define
![{\displaystyle V^{G}=\{v\in V:\rho (s)v=v\,\,\,\forall s\in G\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f15530167193cea6f8a56a1aba84fc390bd5b01b)
In general
is not
–linear. Let
![{\displaystyle Pv:=\int _{G}\rho (s)vds.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3ab13b3baa25c1c6be881f3a4ebf1c97bea876)
The map
is defined as endomorphism on
by having the property
![{\displaystyle \left.\left(\int _{G}\rho (s)vds\right|w\right)=\int _{G}(\rho (s)v|w)ds,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de66d6243cf1849b743aca7823c55fc862468c76)
which is valid for the inner product of the Hilbert space
Then
is
–linear, because of
![{\displaystyle {\begin{aligned}\left.\left(\int _{G}\rho (s)(\rho (t)v)ds\right|w\right)&=\int _{G}\left.\left(\rho \left(tst^{-1}\right)(\rho (t)v)\right|w\right)ds\\&=\int _{G}(\rho (ts)v|w)ds\\&=\int (\rho (t)\rho (s)v|w)ds\\&=\left.\left(\rho (t)\int _{G}\rho (s)vds\right|w\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92bb9cbe0d1978b1af00d305d38713b2b1802426)
where we used the invariance of the Haar measure.
- Proposition. The map
is a projection from
to ![{\displaystyle V^{G}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd165da373724cb9b98f27aff7829f10067e938)
If the representation is finite-dimensional, it is possible to determine the direct sum of the trivial subrepresentation just as in the case of finite groups.
Characters, Schur's lemma and the inner product
Generally, representations of compact groups are investigated on Hilbert- and Banach spaces. In most cases they are not finite-dimensional. Therefore, it is not useful to refer to characters when speaking about representations of compact groups. Nevertheless, in most cases it is possible to restrict the study to the case of finite dimensions:
Since irreducible representations of compact groups are finite-dimensional and unitary (see results from the first subsection), we can define irreducible characters in the same way as it was done for finite groups.
As long as the constructed representations stay finite-dimensional, the characters of the newly constructed representations may be obtained in the same way as for finite groups.
Schur's lemma is also valid for compact groups:
Let
be an irreducible unitary representation of a compact group
Then every bounded operator
satisfying the property
for all
is a scalar multiple of the identity, i.e. there exists
such that
Definition. The formula
![{\displaystyle (\Phi |\Psi )=\int _{G}\Phi (t){\overline {\Psi (t)}}dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/052d18b02e771caf0599249779f373025f5f3456)
defines an inner product on the set of all square integrable functions
of a compact group
Likewise
![{\displaystyle \langle \Phi ,\Psi \rangle =\int _{G}\Phi (t)\Psi (t^{-1})dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9448cb422dc45f1743ca4ed167ad749bc8bfc33d)
defines a bilinear form on
of a compact group
The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid:
- Theorem. Let
and
be the characters of two non-isomorphic irreducible representations
and
respectively. Then the following is valid ![{\displaystyle (\chi |\chi ')=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0eeb36cb8f46aee90ae5860a3311a04f826e0ad)
i.e.
has "norm" ![{\displaystyle 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24)
- Theorem. Let
be a representation of
with character
Suppose
is an irreducible representation of
with character
The number of subrepresentations of
equivalent to
is independent of any given decomposition for
and is equal to the inner product ![{\displaystyle (\chi _{V}|\chi _{W}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69bd49c8551ee69f09aac31236bc8314b4322810)
- Irreducibility Criterion. Let
be the character of the representation
then
is a positive integer. Moreover
if and only if
is irreducible.
Therefore, using the first theorem, the characters of irreducible representations of
form an orthonormal set on
with respect to this inner product.
- Corollary. Every irreducible representation
of
is contained
–times in the left-regular representation.
- Lemma. Let
be a compact group. Then the following statements are equivalent:
is abelian. - All the irreducible representations of
have degree ![{\displaystyle 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24)
- Orthonormal Property. Let
be a group. The non-isomorphic irreducible representations of
form an orthonormal basis in
with respect to this inner product.
As we already know that the non-isomorphic irreducible representations are orthonormal, we only need to verify that they generate
This may be done, by proving that there exists no non-zero square integrable function on
orthogonal to all the irreducible characters.
Just as in the case of finite groups, the number of the irreducible representations up to isomorphism of a group
equals the number of conjugacy classes of
However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information.
The induced representation
If
is a closed subgroup of finite index in a compact group
the definition of the induced representation for finite groups may be adopted.
However, the induced representation can be defined more generally, so that the definition is valid independent of the index of the subgroup
For this purpose let
be a unitary representation of the closed subgroup
The continuous induced representation
is defined as follows:
Let
denote the Hilbert space of all measurable, square integrable functions
with the property
for all
The norm is given by
![{\displaystyle \|\Phi \|_{G}={\text{sup}}_{s\in G}\|\Phi (s)\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01a12dc4451912cf0b7998eb6971c159cb88463b)
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
![{\displaystyle T:{\text{Hom}}_{G}(V_{\rho },I_{H}^{G}(\eta ))\to {\text{Hom}}_{H}(V_{\rho }|_{H},V_{\eta }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63fb06b1f65c570fab1806c1aeb71483dc9a500)
The Frobenius reciprocity transfers, together with the modified definitions of the inner product and of the bilinear form, to compact groups. The theorem now holds for square integrable functions on
instead of class functions, but the subgroup
must be closed.
The Peter-Weyl Theorem
Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements.
- The Peter-Weyl Theorem. Let
be a compact group. For every irreducible representation
of
let
be an orthonormal basis of
We define the matrix coefficients
for
Then we have the following orthonormal basis of
: ![{\displaystyle \left({\sqrt {\dim(\tau )}}\tau _{k,l}\right)_{k,l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e67e7e8ff81e02234abac4e2fa38df40f811020)
We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups:
- The Peter-Weyl Theorem (Second version).[7] There exists a natural
–isomorphism ![{\displaystyle L^{2}(G)\cong _{G\times G}{\widehat {\bigoplus }}_{\tau \in {\widehat {G}}}{\text{End}}(V_{\tau })\cong _{G\times G}{\widehat {\bigoplus }}_{\tau \in {\widehat {G}}}\tau \otimes \tau ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9726bf57b750405d52caabb0b4308587e0ccd5ff)
- in which
is the set of all irreducible representations of
up to isomorphism and
is the representation space corresponding to
More concretely: ![{\displaystyle {\begin{cases}\Phi \mapsto \sum _{\tau \in {\widehat {G}}}\tau (\Phi )\\[5pt]\tau (\Phi )=\int _{G}\Phi (t)\tau (t)dt\in {\text{End}}(V_{\tau })\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a07ad8edebba16703e8df32819e703402b95d9f0)
History
The general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. Later the modular representation theory of Richard Brauer was developed.
See also
Literature
- Bonnafé, Cedric (2010). Representations of SL2(Fq). Algebra and Applications. Vol. 13. Springer. ISBN 9780857291578.
- Bump, Daniel (2004), Lie Groups, Graduate Texts in Mathematics, vol. 225, New York: Springer-Verlag, ISBN 0-387-21154-3
- [1] Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, New York: Springer Verlag, ISBN 0-387-90190-6
- [2] Fulton, William; Harris, Joe: Representation Theory A First Course. Springer-Verlag, New York 1991, ISBN 0-387-97527-6.
- [3] Alperin, J.L.; Bell, Rowen B.: Groups and Representations Springer-Verlag, New York 1995, ISBN 0-387-94525-3.
- [4] Deitmar, Anton: Automorphe Formen Springer-Verlag 2010, ISBN 978-3-642-12389-4, p. 89-93,185-189
- [5] Echterhoff, Siegfried; Deitmar, Anton: Principles of harmonic analysis Springer-Verlag 2009, ISBN 978-0-387-85468-7, p. 127-150
- [6] Lang, Serge: Algebra Springer-Verlag, New York 2002, ISBN 0-387-95385-X, p. 663-729
- [7] Sengupta, Ambar (2012). Representing finite groups: a semisimple introduction. New York. ISBN 9781461412311. OCLC 769756134.
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References
- ^ (Serre 1977, p. 47)
- ^ (Sengupta 2012, p. 62)
- ^ 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.
- ^ Some authors define the character as
, but this definition is not used in this article. - ^ by using the action of G on itself given by
- ^ A proof of this theorem may be found in [1].
- ^ A proof of this theorem and more information regarding the representation theory of compact groups may be found in [5].