Representation theory (group theory)

The representation theory described here is a branch of mathematics that is based on group theory and is a special case of representation theory itself, which deals with representations of algebras .

The basic idea is to represent the elements of a group by transforming certain mathematical objects.

A representation of a group , also a group representation, is a homomorphism of in the automorphism group of a given structure . The group link in corresponds to the sequential execution of automorphisms in : ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \operatorname {Aut} (W)}$ ${\displaystyle W}$ ${\displaystyle G}$ ${\displaystyle W}$

${\displaystyle \rho (gh)=\rho (g)\rho (h)}$

A linear representation is a representation by automorphisms of a vector space . A linear representation is thus a homomorphism of in the general linear group . If a -dimensional vector space is over a body , then the representation accordingly consists of invertible -matrices with coefficients . The vector space dimension is called the degree of representation. ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle \operatorname {GL} (V)}$ ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle n\times n}$ ${\displaystyle K}$ ${\displaystyle n}$

Often the term “representation” is used in the narrower sense of linear representation ; a representation by arbitrary automorphisms is then called realization.

Linear representations make it possible to investigate properties of a group using the means of linear algebra. This is useful because , unlike group theory, linear algebra is a small, closed, and well-understood field.

Representations of finite groups make it possible in molecular physics and crystallography to determine the effects of existing symmetries on measurable properties of a material with the help of a recipe-based calculation.

→ Formally and also according to the name, the permutation representation belongs to the representations of a group defined here: Here the structure is a finite set, its automorphism group thus the set of its bijective self-mappings. The homomorphism is thus a group operation , the linear representations are also special group operations. For permutation representations, which despite the formal context are not objects of investigation of representation theory, see the article Permutation group . ${\displaystyle W}$

More generally, there are well-developed theories for representation theory of finite groups and representation theory of compact groups .

definition

Linear representations

Let be a vector space and a finite group. A linear representation of a finite group is a group homomorphism d. This means that it applies to everyone. One calls the presentation space of Often the term “presentation of ” is also used for the presentation space. The term linear representation is also used for representations of a group in modules instead of vector spaces. We write for the representation of or only if it is clear to which representation the room should belong. ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V)={\text{Aut}}(V),}$ ${\displaystyle \rho (st)=\rho (s)\rho (t)}$ ${\displaystyle s,t\in G.}$ ${\displaystyle V}$ ${\displaystyle G.}$ ${\displaystyle G}$ ${\displaystyle V}$

${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V_{\rho })}$ ${\displaystyle G}$ ${\displaystyle (\rho ,V),}$ ${\displaystyle V}$

In many contexts one restricts oneself 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 , the representation space of which has finite dimensions. The level of representation is the dimension of the display space is often also of the degree of representation used. ${\displaystyle {\text{dim}}(V)<\infty .}$ ${\displaystyle V}$ ${\displaystyle {\text{dim}}(V)=n}$ ${\displaystyle V.}$ ${\displaystyle {\text{dim}}(\rho )}$ ${\displaystyle \rho }$

Examples

A very simple example is the so-called unity representation or trivial representation, which is given by for all A representation of the degree of a group is a homomorphism into the multiplicative group of Since every element has finite order, the values ​​are roots of unity . ${\displaystyle \rho (s)={\text{Id}}}$ ${\displaystyle s\in G.}$
${\displaystyle 1}$ ${\displaystyle G}$ ${\displaystyle \rho \colon G\to {\text{GL}}(\mathbb {C} )=\mathbb {C} ^{\times }}$ ${\displaystyle \mathbb {C} .}$ ${\displaystyle G}$ ${\displaystyle \rho (s)}$

More nontrivial examples:
Let be a linear representation that is not trivial. Then it is determined by your picture and is one of the three following pictures: ${\displaystyle \rho \colon G=\mathbb {Z} /4\mathbb {Z} \to \mathbb {C} ^{\times }}$ ${\displaystyle \rho }$ ${\displaystyle {\overline {1}}\in \mathbb {Z} /4\mathbb {Z} }$

${\displaystyle \rho _{1}({\overline {0}})=1,\,\rho _{1}({\overline {1}})=i,\,\rho _{1}({\overline {2}})=-1,\,\rho _{1}({\overline {3}})=-i.}$

${\displaystyle \rho _{2}({\overline {0}})=1,\,\rho _{2}({\overline {1}})=-1,\,\rho _{2}({\overline {2}})=1,\,\rho _{2}({\overline {3}})=-1.}$

${\displaystyle \rho _{3}({\overline {0}})=1,\,\rho _{3}({\overline {1}})=-i,\,\rho _{3}({\overline {2}})=-1,\,\rho _{3}({\overline {3}})=i.}$

The image set is therefore a nontrivial subgroup of the group that consists of the fourth roots of unity.

Let and be the group homomorphism defined by: ${\displaystyle G=\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \rho \colon G\to {\text{GL}}_{2}(\mathbb {C} )}$

${\displaystyle \rho (0,0)=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right),\,\,\rho (1,0)=\left({\begin{array}{cc}-1&0\\0&-1\end{array}}\right),\,\,\rho (0,1)=\left({\begin{array}{cc}0&1\\1&0\end{array}}\right),\,\,{\text{und}}\,\,\rho (1,1)=\left({\begin{array}{cc}0&-1\\-1&0\end{array}}\right).}$

Then is a linear representation of the degree . ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle 2}$

Let be the cyclic group , i.e. the set with the addition modulo as a group link. ${\displaystyle G}$ ${\displaystyle C_{3}}$ ${\displaystyle \{0,1,2\}}$ ${\displaystyle 3}$

The mapping that assigns powers of the complex number to the group elements is a faithful linear representation of degree . The group property corresponds to the property . The multiplicative group generated by the representation is isomorphic to the group represented . ${\displaystyle \tau \colon G\to \mathbb {C} }$ ${\displaystyle g}$ ${\displaystyle \tau (g)=u^{g}}$ ${\displaystyle u=e^{\frac {2\pi i}{3}}}$ ${\displaystyle 1}$ ${\displaystyle g^{3}=e}$ ${\displaystyle u^{3}=1}$ ${\displaystyle \tau (C_{3})=\{1,u,u^{2}\}}$ ${\displaystyle C_{3}}$

Such an isomorphism is also present in the true linear representation of degree 2, which is given by

${\displaystyle \rho (0)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}},\qquad \rho (1)={\begin{bmatrix}1&0\\0&u\\\end{bmatrix}},\qquad \rho (2)={\begin{bmatrix}1&0\\0&u^{2}\\\end{bmatrix}}.}$

This representation is equivalent to a representation using the following matrices:

${\displaystyle \rho '(0)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}},\qquad \rho '(1)={\begin{bmatrix}u&0\\0&1\\\end{bmatrix}},\qquad \rho '(2)={\begin{bmatrix}u^{2}&0\\0&1\\\end{bmatrix}}.}$

The representations and are reducible: They consist of the direct sum of the representation described above and the untrue representation . ${\displaystyle \rho }$ ${\displaystyle \rho ^{\prime }}$ ${\displaystyle g\to u^{g}}$ ${\displaystyle g\to 1}$

A real representation of this group is obtained by assigning the rotation of the real plane by 120 degrees. This representation is irreducible over the real numbers. If you let the operate as a 120-degree rotation on the complex plane , you get a reducible representation that is isomorphic to the representation considered above . ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \rho }$

glossary

• A representation is called true if the representation homomorphism is injective, i.e. if different group elements are always represented by different transformations. In this case induces an isomorphism between and the image. One can then be understood as a subgroup of the automorphism group of . ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle \rho (G)\subset GL(V).}$ ${\displaystyle G}$ ${\displaystyle V}$

• The trivial representation with (for all ) is generally not faithful. ${\displaystyle \mathbf {1} \colon G\rightarrow \operatorname {GL} _{1}(K)=K^{*}}$ ${\displaystyle g\mapsto 1}$ ${\displaystyle g\in G}$

• Two linear representations are called equivalent if their matrices are similar , that is, represent the same linear representation for different bases; that is, if there is an invertible matrix is such that for all group elements applies: . ${\displaystyle \rho _{1},\rho _{2}}$ ${\displaystyle S}$ ${\displaystyle g}$ ${\displaystyle \rho _{1}(g)=S\rho _{2}(g)S^{-1}}$

• If there is only one representation in a context, one only writes instead of often . ${\displaystyle \rho }$ ${\displaystyle \rho (g)(v)}$ ${\displaystyle gv}$

• Let be a vector space. The representation is called unitary if there is an -invariant, positively definite norm , i.e. i.e., if applies to: for all and for all . ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \rho \colon G\rightarrow \operatorname {GL} (V)}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle \beta =\langle \cdot ,\cdot \rangle }$ ${\displaystyle \beta }$ ${\displaystyle \langle v,w\rangle ={\bigl \langle }\rho (g)(v),\rho (g)(w){\bigr \rangle }=\langle gv,gw\rangle }$ ${\displaystyle g\in G}$ ${\displaystyle v,w\in V}$

• Let be a representation of the group on -vector space . A subspace is called -invariant (more precisely: -invariant) if: for all . ${\displaystyle \rho \colon G\rightarrow GL_{K}(V)}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle U\subseteq V}$ ${\displaystyle G}$ ${\displaystyle \rho }$ ${\displaystyle \rho (g)(U)=gU\subseteq U}$ ${\displaystyle g\in G}$

• The representation (or the representation space ) is called irreducible if there are only the two trivial -invariant subspaces and of . (One of the main tasks of representation theory is the classification of irreducible representations.) Particularly in the non-semi-simple case and when viewed as modules, such representations are also called simple . ${\displaystyle \rho }$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle 0\ (=\{0\})}$ ${\displaystyle V\ (\neq 0)}$ ${\displaystyle V}$

• Is not irreducible, it is called reducible. ${\displaystyle \rho }$ ${\displaystyle \rho }$

• Is a direct sum of irreducible representations of G, then is called completely reducible. In particular, every irreducible representation is completely reducible. ${\displaystyle \rho }$ ${\displaystyle \rho }$

• If it can not be broken down into a nontrivial direct sum of (not necessarily irreducible) representations, it means indivisible, otherwise decomposable. (Note that only in the case of “irreducible” and “indecomposable” are identical according to Maschke's theorem .) ${\displaystyle \rho }$ ${\displaystyle \rho }$ ${\displaystyle \operatorname {char} (K)\nmid \left|G\right|}$

• If there is a representation, then the center of is the set of KG endomorphisms of , that is . Is a matrix representation, that is , then: . According to Schur's lemma , the center for irreducible representations is an oblique body . The converse also applies in the case of a field of characteristic and a finite group , so that it is a skew body if and only if is irreducible. ${\displaystyle \rho \colon G\rightarrow \operatorname {GL} _{K}(V)}$ ${\displaystyle Z(\rho )}$ ${\displaystyle \rho }$ ${\displaystyle V}$ ${\displaystyle Z(\rho )=\operatorname {End} _{KG}(V)=\{f\in \operatorname {End} _{K}(V)\ |\ f\circ \rho (g)=\rho (g)\circ f{\text{ für alle }}g\in G\}}$ ${\displaystyle \rho }$ ${\displaystyle \rho \colon G\rightarrow \operatorname {GL} _{n}(K),\ g\mapsto R_{g}}$ ${\displaystyle Z(\rho )=\{A\in M_{n}(K)\ |\ A\cdot R_{g}=R_{g}\cdot A{\text{ für alle }}g\in G\}}$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle G}$ ${\displaystyle Z(\rho )}$ ${\displaystyle \rho }$

character

→ Main article : Character (math)

definition

The character of the finite-dimensional representation is the function carried out by ${\displaystyle \rho \colon G\to \operatorname {GL} (V)}$ ${\displaystyle \chi _{\rho }\colon G\to K}$

${\displaystyle \chi _{\rho }(g)=\operatorname {tr} {\bigl (}\rho (g){\bigr )}=\sum _{j=1}^{\dim(V)}\rho _{jj}(g)}$

is defined. Here, the matrix elements in an arbitrary (but fixed) of the base . The track is independent of the base. ${\displaystyle \rho _{jj}}$ ${\displaystyle V}$ ${\displaystyle \operatorname {tr} }$

properties

• For a finite group , two representations and are equivalent if it holds and the basic field has the characteristic . ${\displaystyle G}$ ${\displaystyle \rho }$ ${\displaystyle \rho '}$ ${\displaystyle \chi _{\rho }=\chi _{\rho '}}$ ${\displaystyle 0}$
• ${\displaystyle \chi (g)=\chi (hgh^{-1})}$because . Therefore is constant on the conjugation classes. ${\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}$ ${\displaystyle \chi }$
• ${\displaystyle \chi (1_{G})=\dim(V)}$, can be seen directly from the track.
• ${\displaystyle \chi _{\rho \oplus \rho '}=\chi _{\rho }+\chi _{\rho '}}$

With the help of characters, it is possible to check whether a representation is irreducible: A representation of a finite group over an algebraically closed field of the characteristic is irreducible if and only if applies. Here the unitary scalar product of two functions is defined by . (In this case , you can also replace the term in this formula with .) ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle (\chi ,\chi )=1}$ ${\displaystyle (u,v)}$ ${\displaystyle u,v\colon G\to K}$ ${\displaystyle \textstyle (u,v)={\frac {1}{\left|G\right|}}\sum _{g\in G}u\left(g^{-1}\right)v\left(g\right)}$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle u\left(g^{-1}\right)}$ ${\displaystyle {\overline {u\left(g\right)}}}$

Completely reducible representations of finite groups break down into irreducible representations and can thus be "reduced". The representations can be deduced from the characters; one can set up the character table of a representation and use certain orthogonality relations of the unitary scalar products formed with the row or column vectors of these tables.

application

An application of the concept of reducing a “product” (better: tensor product) of two unnecessarily different representations of the same group yields the Clebsch-Gordan coefficients of angular momentum physics, which are important in quantum mechanics .

Images between representations, equivalence of representations

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

${\displaystyle T\colon V_{\rho }\to V_{\pi },}$

so that applies to all : Such a mapping is also called -linear mapping. One can define the core , the image and the coke of standard. These are again modules and thus again provide representations of via the relationship from the previous section ${\displaystyle s\in G}$ ${\displaystyle \pi (s)\circ T=T\circ \rho (s).}$
${\displaystyle G}$ ${\displaystyle T}$ ${\displaystyle G}$ ${\displaystyle G.}$

Two representations are called equivalent or isomorphic if there is a -linear vector space isomorphism between the representation spaces; d. i.e., if there is a bijective linear mapping with . In particular, equivalent representations have the same degree. ${\displaystyle (\rho ,V_{\rho }),(\pi ,V_{\pi })}$ ${\displaystyle G}$ ${\displaystyle T\colon V_{\rho }\to V_{\pi }}$ ${\displaystyle T\circ \rho (s)=\pi (s)\circ T\,\,\,\forall \,s\in G}$

Taxonomy

Representations can be classified according to two aspects: (1) according to the structure of the target set on which the representations act; and (2) according to the structure of the group shown. ${\displaystyle W}$

Classification according to target quantities

A set- theoretical representation is a homomorphism of the group to be represented on the permutation group of any set ; see also Cayley's theorem . ${\displaystyle \operatorname {Sym} (M)}$ ${\displaystyle M}$

A linear representation is characterized by its dimension and by the body . In addition to the complex and real numbers, the finite and -adic fields come into consideration here. ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle p}$

A linear representation of a finite group over a field of the characteristic is called a modular representation if it is a divisor of the group order. ${\displaystyle p>0}$ ${\displaystyle p}$

Representations in subgroups of the general linear group are characterized by the fact that they contain certain structures of the vector space . For example, a unitary representation, i.e. a representation in the unitary group , receives the scalar product , see also Hilbert space representation . ${\displaystyle \operatorname {GL} (V)}$ ${\displaystyle V}$ ${\displaystyle \operatorname {U} (V)}$

Classification according to the group shown

The simplest case is the representation of a finite group .

Many results in representation theory of finite groups are obtained by averaging over the group. These results can be transferred to infinite groups, provided that the topological prerequisites are given to define an integral . This is possible in locally compact groups by means of the hair measure . The resulting theory plays a central role in harmonic analysis . The Pontryagin duality describes this theory in the special case of Abelian groups as a generalized Fourier transform .

Many important Lie groups are compact , so that the results mentioned are transferable. Representation theory is critical to the applications of these Lie groups in physics and chemistry.

There is no complete theory of representation for non-compact groups. A comprehensive theory has been worked out for semi-simple Lie groups. There is no comparable classification for the complementary resolvable Lie groups.

literature

• Jean-Pierre Serre: Linear Representations of Finite Groups. Springer-Verlag, New York 1977, ISBN 3-540-90190-6 .
• William Fulton, Joe Harris: Representation theory. A first course. Springer-Verlag, New York, 1991, ISBN 0-387-97527-6 .

Web links

• E. Kowalski: Representation Theory. ETH Zurich (PDF; 1.6 MB).