Group representation

The representation theory described here is a branch of mathematics that builds on the theory of groups and a special case of actual representation theory, which deals with representations of algebras.

The basic idea is to present the elements of a group of transformations of certain mathematical objects.

A representation ρ of a group G, and group representation is a homomorphism of G into the automorphism group Aut ( W ) of a given structure W. The group link in G corresponds to the concatenation run automorphisms in W: ρ ( gh ) = ρ ( g) ρ ( h).

A linear representation is a representation of a vector space by automorphisms V. A linear representation is thus a homomorphism of G in the general linear group GL (V). Where V is an n- dimensional vector space over a field K, then there is a depiction of correspondingly invertible NxN matrices with coefficients in the vector space K. The dimension n is the degree of representation.

Often, the term representation is used in the narrower sense of linear representation; a representation by any automorphisms then called realization.

→ Formal and after the designation, the permutation is one of the representations of a group defined here: Here the structure W is a finite set whose automorphism ie the set of its bijective self-maps. Thus the homomorphism is a group operation, the linear representations are special operations group. See to permutation that no objects of study of the representation theory, in spite of the formal context, the article permutation.

• 5.1 Classification by target quantities
• 5.2 Classification by group shown

Glossary

• A representation is called faithful if the Darstellungshomomorphismus is injective, ie if several group members are always represented by various transformations.

• The trivial representation is not faithful in general.

• Two linear representations ρ1, ρ2 are called equivalent if their matrices are similar, so the same linear mapping for different bases represent. That is, when there is an invertible matrix S such that g is valid for all the group members: ρ1 ( g) = S ρ2 ( g) S -1.

• Occurs in a context only a representation on, one writes instead often only

• Let V be a vector space. The presentation is called unitary if V on a G -invariant positive definite standard exists, ie, if for the following applies:

• Let be a representation of the group G on the K- vector space V. A subspace is called G- invariant (more precisely - invariant ) if the following holds:

• The display (or the display area V) is irreducible if there is only the two trivial invariant subspaces and G- V. ( A principal object of representation theory is the classification of irreducible representations. ) In particular, in the non - semisimple case, and in the approach as modules such representations are also called simple.
• Is not irreducible, so is called reducible.
• Is a direct sum of irreducible representations of G, so called completely reducible. In particular, each irreducible representation is completely reducible.
• If it is impossible in a non-trivial direct sum of (not necessarily irreducible ) representations decompose, so called indecomposable otherwise dismantled. ( Note that only in the case by the theorem of Maschke are irreducible and indecomposable identical. )

• Is a representation, it is referred to as the center of the set of KG- endomorphism of V, that is:

Applications

Linear representations make it possible to investigate properties of a group with the means of linear algebra; this is useful because the linear algebra, as opposed to group theory, a small, self and well -understood area is.

Representations of finite groups enable it in molecular physics and crystallography to determine the effects of existing symmetries on measurable properties of a material by means of a prescription moderate calculus.

Example

Let G be the cyclic group C3, ie the set { 0,1,2 } with addition modulo 3 as a group link.

The mapping τ: G → C, the τ of the group elements g potencies ( g) = ug of the complex number u = exp ( 2πi / 3) assigns, is a faithful linear representation of degree 1, the group property g3 = e corresponds to the property u3 = 1, the multiplicative group generated by the representation τ (C3 ) = { 1, u, u2 } is isomorphic to the group represented C3.

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

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

The representations ρ and ρ ' are reducible: they consist of the direct sum of the representation described above g → ug and the unfaithful representation g → 1

A real representation of this group are obtained by assigning one of the rotation of the real plane by 120 degrees. This representation is irreducible over the reals. Excluding the one corresponding to 120 degrees of rotation on the complex plane operate, we obtain a reducible representation which is isomorphic to the representation considered above.

Character

The character of the finite representation is the function defined by

Is defined. The matrix elements in an arbitrary (but fixed ) basis of V. The track is based are independent. Apply the following properties:

• For a finite group G are two representations and already then equivalent if true and the body has the characteristic 0.
• Because. Therefore, is constant on the conjugacy classes.
• Directly from track visible

Use of characters is possible to check whether a representation is irreducible: A representation of a finite group G over an algebraically closed field K of characteristic 0 is irreducible if and only if the following holds. Here, the unitary scalar product of two functions is defined by. ( In case you can replace in this formula, the term also. )

Completely reducible representations of finite groups decompose into irreducible representations and can therefore " ausreduziert " are. Here you can tap into the representations of the characters; you can to set up the character table of a representation and exploit certain orthogonality of the unitary scalar products formed with the row or column vectors of these panels.

Application

An application of the concept of Ausreduzierung a " product " (better: tensor product ) of two different representations of the same group not necessary gives the Clebsch -Gordan coefficients of the angular momentum physics that are important in quantum mechanics.

Taxonomy

Representations can be classified according to two criteria: ( 1) according to the structure of the target set W, which act on the representations; and (2) on the structure of the group represented.

Classification by target quantities

A set-theoretic representation is a homomorphism of the group to be displayed on the permutation group Sym ( M) of an arbitrary set M; see also the theorem of Cayley.

A linear representation of n is its sheer size and characterized by the body K. In addition to the complex and real numbers here come the finite and p- adic field into account.

A linear representation of a finite group over a field of characteristic is called a modular representation, if a divisor of the group order.

Representations in subgroups of the general linear group GL (V ) are characterized by the fact that they receive certain structures of the vector space V. For example, replaced by a unitary display, that is, a representation in said unitary group U (V), the dot product, also see Hilbert space representation.

View the illustrated group

The simplest case is the representation of a finite group.

Many results in the representation theory of finite groups are obtained by averaging over the group. These results can be applied to infinite groups, provided that the topological conditions are given in order to define an integral. This is possible by means of the hair - measure in locally compact groups. The resulting theory plays a central role in the 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 provided are transferable; the representation theory is crucial to the applications of these Lie groups in physics and chemistry.

For non- compact groups, there is not a closed representation theory. A comprehensive theory has been developed for semi - simple Lie groups. For the complementary resolvable Lie groups, there is no comparable classification.