Representation theory of finite groups

The representation theory of finite groups is a branch of mathematics , studied in which one such groups operate on existing structures.

Above all, one looks at the operations of groups on vector spaces (linear representations). However, the operations of groups on other groups or on sets ( permutation representation ) are also considered.

In this article, apart from marked exceptions, only finite groups are considered. We also limit ourselves to vector spaces over basic fields of the characteristic. The theory of the algebraically closed fields of the characteristic is closed, that is, if one theory is valid for an algebraically closed field of the characteristic , it also applies to all others. In the following we can therefore consider vector spaces over without restriction . ${\displaystyle 0.}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle \mathbb {C} }$

Representation theory is used in many areas of mathematics, as well as in quantum chemistry and physics. Representation theory is used, among other things, in algebra to examine the structure of groups, in harmonic analysis and in number theory . For example, representation theory is used in the modern approach to better understand automorphic forms.

Large parts of the representation theory of finite groups can be generalized to the representation theory of compact groups .

history

Characters of finite Abelian groups have been a tool used in number theory since the 18th century, but Frobenius was the first to extend this concept to non- Abelian groups in 1896. The theory of the characters of symmetrical and alternating groups was worked out by Frobenius and Young around 1900 .

Burnside and Schur formulated Frobenius' character theory on the basis of matrix representations instead of characters. Burnside proved that every finite-dimensional representation of a finite group leaves a scalar product invariant and thus obtained a simpler proof of the (already known) decomposability into irreducible representations. Schur proved the lemma named after him and the orthogonality relations.

It was Emmy Noether who gave the customary definition of representations using linear mappings of a vector space.

Schur observed in 1924 that one can use invariant integration to extend the representation theory of finite groups to compact groups . Weyl then developed the representation theory of compact, connected Lie groups .

Definitions

Linear representations

→ Main article : Representation (group)

Let be a vector space and a group. One representation of is a group homomorphism . One calls the representation space of ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V)={\text{Aut}}(V)}$ ${\displaystyle V}$ ${\displaystyle G.}$

We write for the representation of or only if it is clear to which representation the room should belong. ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V_{\rho })}$ ${\displaystyle G}$ ${\displaystyle (\rho ,V),}$ ${\displaystyle V}$

Except for the last chapter, this article is limited to the case Since in most cases one is only interested in a finite number of vectors , one can restrict oneself to a partial representation whose representation space has finite dimensions. The degree of presentation is the dimension of the presentation space ${\displaystyle {\text{dim}}(V)<\infty .}$ ${\displaystyle V}$ ${\displaystyle {\text{dim}}(V)=n}$ ${\displaystyle V.}$

In this article only representations on complex vector spaces are considered, so special classes of such representations are real representations and quaternionic representations . ${\displaystyle K=\mathbb {C} .}$

Examples of representations of finite groups are the permutation representation and the left and right regular representation .

Figures between presentations

→ Main article : "Images between representations, equivalence of representations" in the article Representation theory (group theory)

A mapping between two representations of the same group is a -linear mapping ${\displaystyle (\rho ,V_{\rho }),\,(\pi ,V_{\pi })}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle T\colon V_{\rho }\to V_{\pi }.}$

Two representations are called equivalent or isomorphic if there is a -linear vector space isomorphism between the representation spaces. That is, if there is a bijective linear mapping such that ${\displaystyle (\rho ,V_{\rho }),(\pi ,V_{\pi })}$ ${\displaystyle G}$ ${\displaystyle T\colon V_{\rho }\to V_{\pi }}$ ${\displaystyle T\circ \rho (s)=\pi (s)\circ T\,\,\,\forall \,s\in G.}$

Let be a linear representation of If an -invariant subspace of is, that is, for all the restriction is an isomorphism on Since restriction and group homomorphism are compatible, this construction provides a representation of on This is called a partial representation or subrepresentation of ${\displaystyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle G.}$ ${\displaystyle W}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \rho (s)w\in W}$ ${\displaystyle s\in G,w\in W,}$ ${\displaystyle \rho (s)|_{W}}$ ${\displaystyle W.}$ ${\displaystyle G}$ ${\displaystyle W.}$ ${\displaystyle V.}$

Representation ring, modules and the convolutionalgebra

→ Main article : Presentation ring

Let be a group of finite order and a commutative ring . With we denote the group algebra of over This algebra is free and has a base indexed with the group elements. Mostly the base is identified with . Each element can then be written as unambiguous. The multiplication in continues that in distributive. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K[G]}$ ${\displaystyle G}$ ${\displaystyle K.}$ ${\displaystyle G}$ ${\displaystyle f\in K[G]}$ ${\displaystyle \textstyle f=\sum _{s\in G}a_{s}s}$ ${\displaystyle a_{s}\in K.}$ ${\displaystyle K[G]}$ ${\displaystyle G}$

The representation ring of is defined as the Abelian group ${\displaystyle G}$

${\displaystyle R(G)=\{\sum _{j=1}^{m}a_{j}\tau _{j}|a_{j}\in \mathbb {Z} ,\tau _{j}{\mbox{ irreduzible Darstellungen von }}G{\mbox{ bis auf Isomorphie}}\},}$

which, when multiplied by the tensor product, becomes a ring. The elements of are called virtual representations. ${\displaystyle R(G)}$

Now let one - module and is a linear representation of in For elements and defining through linear extension provides this on the structure of a left module. Conversely, a linear representation of in can be derived from a links module . Therefore, the two terms can be used interchangeably. It is also true that the left module, which is given by itself, corresponds to the left regular representation, just as the right module corresponds to the right regular representation. ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle G}$ ${\displaystyle V.}$ ${\displaystyle s\in G}$ ${\displaystyle v\in V}$ ${\displaystyle sv=\rho (s)v.}$ ${\displaystyle V}$ ${\displaystyle K[G]}$ ${\displaystyle K[G]}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle V}$
${\displaystyle K=\mathbb {C} }$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle \mathbb {C} [G]}$

For a group with the amount of addition and scalar multiplication is a vector space, isomorphic to The folding is then an algebra , the convolution algebra . ${\displaystyle G}$ ${\displaystyle g:={\text{ord}}(G),}$ ${\displaystyle L^{1}(G):=\{f\colon G\to \mathbb {C} \}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \textstyle \mathbb {C} ^{g}.}$ ${\displaystyle \textstyle f*h(s)=\sum _{t\in G}f(t)h(t^{-1}s)}$ ${\displaystyle \textstyle L^{1}(G)}$

Constructions of representations

• Dual representation
• Direct sum of representations
• Tensor product of representations
• Symmetrical square
• Alternating square

Decomposition of representations

→ Main article : Canonical decomposition

Basic concepts

A representation is called irreducible or simple if there is no real -invariant subspace. In the language of group algebra, the irreducible representations correspond to the simple modules. One can show that the number of irreducible representations of a finite group (or the number of simple -modules) corresponds to the number of conjugation classes of . ${\displaystyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} [G]}$
${\displaystyle G}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle G}$

If a representation can be written as a direct sum of irreducible representations, it is called semi-simple or fully reducible. This is a definition analogous to the fact that an algebra that can be written as a direct sum of simple subalgebras is called semi-simple.

A representation is called isotypic if it is a direct sum of isomorphic irreducible representations. Let be any representation of the group. Let be an irreducible representation of, then the -isotype of is defined as the sum of all irreducible sub-representations of which are too isomorphic. ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle G.}$ ${\displaystyle \tau }$ ${\displaystyle G,}$ ${\displaystyle \tau }$ ${\displaystyle V_{\rho }(\tau )}$ ${\displaystyle G}$ ${\displaystyle V,}$ ${\displaystyle \tau }$

Unitarizability

Via we can equip every vector space with a scalar product . A representation of a group in a vector space with a scalar product is called unitary if it is unitary for each (ie, in particular, each is diagonalizable ). A representation is unitary with respect to a given scalar product if and only if the scalar product is invariant under the operation of induced by the representation . For representations of finite groups, a given scalar product can always be replaced by an invariant one, in which one can replace with So we can assume without restriction that all representations considered below are unitary. ${\displaystyle \mathbb {C} }$ ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle \rho (s)}$ ${\displaystyle s\in G}$ ${\displaystyle \rho (s)}$
${\displaystyle G}$
${\displaystyle (\cdot |\cdot )}$ ${\displaystyle (v|u)}$ ${\displaystyle \textstyle \sum _{t\in G}(\rho (t)v|\rho (t)u).}$

Semi-simplicity

→ Main article : Maschke's theorem

In order to be able to understand representations more easily, one would like to break down the representation space into the direct sum of simpler partial representations. The following results are obtained for representations of finite groups over a field of the characteristic . ${\displaystyle 0}$

• Let be a linear representation and be an -invariant subspace of Then the complement of in exists and is also -invariant. ${\displaystyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle W}$ ${\displaystyle G}$ ${\displaystyle V.}$ ${\displaystyle \textstyle W^{0}}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle \textstyle W^{0}}$ ${\displaystyle G}$

A partial representation and its complement clearly define the representation.

• Any linear representation of compact groups is a direct sum of irreducible representations.

In the formulation of the modules this means: If so, the group algebra is semi-simple, ie it is the direct sum of simple algebras. This breakdown is ambiguous. However, the number of partial representations that occur, which are isomorphic to a given irreducible representation, does not depend on the decomposition chosen. ${\displaystyle K[G]}$ ${\displaystyle {\text{char}}(K)=0,}$ ${\displaystyle K[G]}$

The canonical decomposition

In order to get a clear decomposition, one summarizes all isomorphic irreducible partial representations. So the representation space is broken down into the direct sum of its isotypes. This decomposition is clear. It is called the canonical division.
Let be the family of all irreducible representations of a group except for isomorphism. Let Let be a representation of and the set of isotypes of The projection to the canonical decomposition is given by ${\displaystyle (\tau _{j})_{j\in I}}$ ${\displaystyle G}$ ${\displaystyle g={\text{ord}}(G).}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle \{V(\tau _{j})|j\in I\}}$ ${\displaystyle V.}$ ${\displaystyle p_{j}\colon V\to V(\tau _{j})}$

${\displaystyle p_{j}={\frac {n_{j}}{g}}\sum _{t\in G}{\overline {\chi _{\tau _{j}}(t)}}\rho (t),}$

where and is the associated character. ${\displaystyle n_{j}={\text{dim}}(\tau _{j})}$ ${\displaystyle \chi _{\tau _{j}}}$ ${\displaystyle \tau _{j}}$

In the following we see how one can determine the isotype for the trivial representation:

Projection formula: For each representation of a group with define the mapping is a projection from to In general, is non- linear. Set Then a linear map, as for all ${\displaystyle (\rho ,V)}$ ${\displaystyle G}$ ${\displaystyle g={\text{ord}}(G)}$ ${\displaystyle V^{G}=\{v\in V\mid \rho (s)v=v\,\,\,\,\forall s\in G\}.}$ ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle V^{G}.}$
${\displaystyle \rho (s)\colon V\to V}$ ${\displaystyle G}$ ${\displaystyle \textstyle P:={\frac {1}{g}}\sum _{s\in G}\rho (s)\in {\text{End}}(V).}$ ${\displaystyle P}$ ${\displaystyle G}$ ${\displaystyle \textstyle \sum _{s\in G}\rho (s)=\sum _{s\in G}\rho (tst^{-1})}$ ${\displaystyle t\in G.}$

The number of times the trivial representation occurs in is given by the trace of This follows, since a projection can only have the eigenvalues and and the eigenspace to the eigenvalue is the image of the projection. Since the trace is the sum of the eigenvalues, one thus obtains ${\displaystyle V}$ ${\displaystyle P.}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 1}$

${\displaystyle {\text{dim}}(V(1))={\text{dim}}(V^{G})=Tr(P)={\frac {1}{g}}\sum _{s\in G}\chi _{V}(s),}$

where denotes the isotype for the trivial representation and Let be a nontrivial irreducible representation of then the isotype for the trivial representation of the null space. That is, it applies ${\displaystyle V(1)}$ ${\displaystyle g={\text{ord}}(G).}$
${\displaystyle V_{\pi }}$ ${\displaystyle G,}$ ${\displaystyle \pi }$

${\displaystyle P={\frac {1}{g}}\sum _{s\in G}\pi (s)=0.}$

Let be an orthonormal basis of Then: ${\displaystyle e_{1},\dotsc ,e_{n}}$ ${\displaystyle V_{\pi }.}$

${\displaystyle \sum _{s\in G}{\text{Tr}}(\pi (s))=\sum _{s\in G}\sum _{j=1}^{n}\langle \pi (s)e_{j},e_{j}\rangle =\sum _{j=1}^{n}\langle \sum _{s\in G}\pi (s)e_{j},e_{j}\rangle =0.}$

So it is true for a non-trivial irreducible representation ${\displaystyle V:}$

${\displaystyle \sum _{s\in G}\chi _{V}(s)=0.}$

That the above theorems on the decomposition for infinite groups are no longer necessarily valid, should be illustrated here with an example: Let Then with the matrix multiplication, a group of infinite power is. The group operates on by matrix-vector multiplication. We consider the representation for all. The subspace is an -invariant subspace. But there is no -invariant complement for this subspace . The assumption that such a complement exists leads to the contradictory result that every matrix can be diagonalized over . That is, if we consider infinite groups, the case may arise that a representation is not irreducible, but nevertheless does not break down into the direct sum of irreducible partial representations. ${\displaystyle \textstyle G=\{A\in {\text{GL}}_{2}(\mathbb {C} )|\,A\,\,{\text{ ist}}{\text{ obere}}{\text{ Dreiecksmatrix}}\}.}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \rho (A)=A}$ ${\displaystyle A\in G.}$ ${\displaystyle \mathbb {C} e_{1}}$ ${\displaystyle G}$
${\displaystyle G}$ ${\displaystyle \mathbb {C} }$

Lemma from Schur

→ Main article : Schur's Lemma

Be and two irreducible linear representations. Let it be a linear mapping such that for all then: ${\displaystyle \rho _{1}\colon G\to {\text{GL}}(V_{\rho _{1}})}$ ${\displaystyle \rho _{2}\colon G\to {\text{GL}}(V_{\rho _{2}})}$ ${\displaystyle F\colon V_{\rho _{1}}\to V_{\rho _{2}}}$ ${\displaystyle \rho _{2}(s)\circ F=F\circ \rho _{1}(s)}$ ${\displaystyle s\in G.}$

• If and are not isomorphic, is ${\displaystyle \rho _{1}}$ ${\displaystyle \rho _{2}}$ ${\displaystyle F=0.}$
• If and so is a homothety (ie, for a ). ${\displaystyle V_{\rho _{1}}=V_{\rho _{2}}}$ ${\displaystyle \rho _{1}=\rho _{2},}$ ${\displaystyle F}$ ${\displaystyle F=\lambda {\text{Id}}}$ ${\displaystyle \lambda \in \mathbb {C} }$

Character theory

→ Main article : Character (math)

An essential tool in representation theory of finite groups is character theory. Let be a linear representation of the finite group in the vector space. Define the mapping by where denotes the trace of the linear mapping . The complex-valued function thus obtained is called the character of the representation. Sometimes the character of a representation is also defined as where the degree of representation. However, this definition is not used in this article. Based on the definition, you can immediately see that isomorphic representations have the same character. ${\displaystyle \textstyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle \textstyle G}$ ${\displaystyle \textstyle V.}$ ${\displaystyle \chi _{\rho }}$ ${\displaystyle \textstyle \chi _{\rho }(s)={\text{Tr}}(\rho (s)),}$ ${\displaystyle {\text{Tr}}(\rho (s))}$ ${\displaystyle \rho (s)}$ ${\displaystyle \textstyle \chi _{\rho }}$ ${\displaystyle \textstyle \rho .}$
${\displaystyle \textstyle \rho }$ ${\displaystyle \textstyle \chi (s)={\text{dim}}(\rho ){\text{Tr}}(\rho (s)),}$ ${\displaystyle \textstyle {\text{dim}}(\rho )}$

A scalar product can be defined on the set of all characters of a finite group : ${\displaystyle G}$

${\displaystyle (f|h)_{G}={\frac {1}{|G|}}\sum _{t\in G}f(t){\overline {h(t)}}.}$

For two modules we define where is the vector space of all -linear mappings. This form is bilinear with respect to the direct sum. ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle V_{1},V_{2}}$ ${\displaystyle \langle V_{1},V_{2}\rangle _{G}:={\text{dim}}({\text{Hom}}^{G}(V_{1},V_{2})),}$ ${\displaystyle {\text{Hom}}^{G}(V_{1},V_{2})}$ ${\displaystyle G}$

Orthogonality relations

→ Main article : Orthogonality relations

This scalar product makes it possible to obtain important results with regard to the decomposition and irreducibility of representations.

Theorem: If the characters of two non-isomorphic irreducible representations then hold ${\displaystyle \chi ,\chi '}$ ${\displaystyle V,V',}$

• ${\displaystyle (\chi |\chi ')=0.}$
• ${\displaystyle (\chi |\chi )=1,}$d. h., has "norm" ${\displaystyle \chi }$ ${\displaystyle 1.}$

Corollary: If the characters are then: ${\displaystyle \chi _{1},\chi _{2}}$ ${\displaystyle V_{1},V_{2}}$ ${\displaystyle \langle \chi _{1},\chi _{2}\rangle _{G}=(\chi _{1}|\chi _{2})_{G}=\langle V_{1},V_{2}\rangle _{G}.}$

Theorem: Let be a linear representation of with character Let the irreducible apply . Let now be an irreducible representation of with character Then the following applies: The number of partial representations that are too equivalent does not depend on the given decomposition and corresponds to the scalar product , i.e. the isotype of is independent of the choice of decomposition and it applies ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle \xi .}$ ${\displaystyle V=W_{1}\oplus \cdots \oplus W_{k},}$ ${\displaystyle W_{j}}$ ${\displaystyle (\tau ,W)}$ ${\displaystyle G}$ ${\displaystyle \chi .}$
${\displaystyle W_{j},}$ ${\displaystyle W}$ ${\displaystyle (\xi |\chi ).}$
${\displaystyle \tau }$ ${\displaystyle V(\tau )}$ ${\displaystyle V}$

${\displaystyle (\xi |\chi )={\frac {{\text{dim}}(V(\tau ))}{{\text{dim}}(\tau )}}=\langle V,W\rangle }$

and thus

${\displaystyle {\text{dim}}(V(\tau ))={\text{dim}}(\tau )(\xi |\chi ).}$

Corollary: Two representations with the same character are isomorphic. In other words, each representation is determined by its character.

Irreducibility criterion: Let the character of a representation then is and it applies exactly when is irreducible. ${\displaystyle \chi }$ ${\displaystyle V,}$ ${\displaystyle (\chi |\chi )\in \mathbb {N} _{0}}$ ${\displaystyle (\chi |\chi )=1}$ ${\displaystyle V}$

The characters of irreducible representations of thus form an orthonormal system with respect to this scalar product to the equivalent of the orthonormal property: The number of all irreducible representations of a group except for isomorphism corresponds exactly to the number of all conjugation classes of ${\displaystyle G}$ ${\displaystyle \mathbb {C} _{\text{class}}(G).}$ ${\displaystyle G}$ ${\displaystyle G.}$

Corollary: Let be a vector space with Every irreducible representation of is -mal in the regular representation . That means, for the regular representation of : where describes the set of all irreducible representations of that are pairwise not isomorphic to one another. In words of group algebra we get as algebras. ${\displaystyle V}$ ${\displaystyle {\text{dim}}(V)=n.}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle \textstyle R\cong \oplus (W_{j})^{\oplus {\text{dim}}(W_{j})},}$ ${\displaystyle \{W_{j}|j\in I\}}$ ${\displaystyle G}$
${\displaystyle \mathbb {C} [G]\cong \oplus _{j}{\text{End}}(W_{j})}$

Other applications of this theory include the Fourier inversion formula and Plancherel's formula.

Induced representations

→ Main article : Induced representation

Using the restriction you can obtain a representation of a subset of a representation of a group:
Be a subgroup of the group For an account of is the restriction of the subgroup ${\displaystyle H}$ ${\displaystyle G.}$ ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle {\text{Res}}_{H}(\rho )}$ ${\displaystyle \rho }$ ${\displaystyle H.}$

The question that now arises is that of the reverse process. Can one get a representation of the whole group from a given representation of a subgroup? It can be seen that the induced representation defined below delivers exactly what we are looking for. However, this construction is not inverse, but adjoint to the restriction.

Definition: Let be a linear representation of Let be a subgroup and the constraint. Let be a partial representation of Write for this representation. Let the vector space depend only on the left minor class of . Let be a representative system of then is a partial representation of ${\displaystyle \rho \colon G\to {\text{GL}}(V_{\rho })}$ ${\displaystyle G.}$ ${\displaystyle H}$ ${\displaystyle \rho |_{H}}$ ${\displaystyle W}$ ${\displaystyle \rho _{H}.}$ ${\displaystyle \theta \colon H\to {\text{GL}}(W)}$ ${\displaystyle s\in G,}$ ${\displaystyle \rho (s)(W)}$ ${\displaystyle sH}$ ${\displaystyle s}$ ${\displaystyle R}$ ${\displaystyle G/H,}$ ${\displaystyle \textstyle \sum _{r\in R}\rho (r)(W)}$ ${\displaystyle V_{\rho }.}$

A representation of in is called induced by the representation of in if There is a system of representatives of as above and for each We write for the representation of induced representation of The induced representation exists and is unique. ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle V_{\rho }}$ ${\displaystyle \theta }$ ${\displaystyle H}$ ${\displaystyle W,}$ ${\displaystyle \textstyle V_{\rho }=\bigoplus _{r\in R}W_{r}.}$ ${\displaystyle R}$ ${\displaystyle G/H}$ ${\displaystyle W_{r}=\rho (s)(W)}$ ${\displaystyle s\in rH.}$ ${\displaystyle {\text{Ind}}_{H}^{G}(\theta )}$ ${\displaystyle \theta }$ ${\displaystyle H}$ ${\displaystyle G.}$

An important property in representation theory of finite groups is Frobenius reciprocity . On the one hand, it tells us that the images and are adjoint to one another. If, on the other hand, we consider an irreducible representation of and be an irreducible representation of, then with Frobenius reciprocity we also get that in is contained as often as in ${\displaystyle {\text{Res}}}$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle W}$ ${\displaystyle H}$ ${\displaystyle V}$ ${\displaystyle G,}$ ${\displaystyle W}$ ${\displaystyle {\text{Res}}(V)}$ ${\displaystyle {\text{Ind}}(W)}$ ${\displaystyle V.}$

With the Mackey criterion the irreducibility of induced representations can be checked.

Important sentences

• Maschke's theorem : Can be decomposed into irreducible representations
• Orthogonality relations : orthonormality of irreducible characters
• Frobenius reciprocity
• Artin's theorem : Decomposability as a rational linear combination of characters of cyclical groups
• Brauer's theorem : Decomposability as an integral linear combination of characters of elementary groups

literature

• Jean-Pierre Serre; Linear representations of finite groups. Translated into German from French and edited by Günter Eisenreich. Akademie-Verlag, Berlin, 1972.

Web links

• Webb: A course in finite group representation theory.

Individual evidence

1. ↑ Frobenius: About group characters. Meeting reports of the Royal Prussian Academy of Sciences in Berlin (1896), 985-1021; in Gesammelte Abhandlungen, Volume III Springer-Verlag, New York, 1968, 1-37.
2. ^ Anthony W. Knapp: Group representations and harmonic analysis from Euler to Langlands. Notices of the American Mathematical Society 43 (1996).