Irreducible representation

In Mathematical, a irreducible representation is a concept used within the framework of the theory of the representation of a group.

A irreducible representation is a representation which does not admit that itself and the null representation like underrepresentation.

This concept is important because the Théorème of Maschke shows that, in many cases, a representation is direct Somme of irreducible representations .

Definitions and examples

Definitions

Subsequently of the article, G indicates a group and ( V , ρ) a linear representation of G on a body K .

* a representation (V, ρ) is known as irreducible if and only if the only stable subspaces are V and the null vector.

* a character of a representation is known as irreducible if and only if the associated representation is.

The theory of the representations is also expressed in term of '' G '' module. V naturally lays out of a structure of G module, in this context, the definition takes the following form:

* a representation (V, ρ) is known as irreducible if and only if V is simple as a G module.

* a representation (V, ρ) is known as isotypic if and only if the only stable irreducible subspaces different from the null representation are isomorphous two to two.

Examples

All representations of dimension a is irreducible.

There exists only one irreducible and faithful representation symmetrical Groupe of index three. The articles Representations of the symmetrical group of index three and Représentations of the symmetrical group of index four contain an exhaustive analysis of the irreducible representations of these groups.

If V indicates a vector Space real of Dimension two and G the group of the Isométrie S linear of V , then the identity of G is an irreducible representation.

Theorem of Maschke

See also: Theorem of Maschke

The theorem of Maschke indicates that any irreducible subspace of the representation ( V , ρ) is direct factor, i.e. that it has a additional Sous-espace stable.

This theorem applies at least in two important cases:

* If the group is finished and if the characteristic of K is either null or first with the order of the group.
* If the group is topological and has a Mesure of Haar.

In this case, V lays out of a semi-simple structure of Module. Any representation of finished degree of G is then direct sum of representations irreducible.

the demonstrations are given in the associated article .

Case of a finished group

One supposes in this paragraph of G is a group finished G and that the characteristic of K is either null or first with the order of the group. The theorem of Maschke applies then. ( W , σ) an irreducible representation of G of degree D indicates here. It is supposed finally that the Polynôme X g - 1 is divided in K .

Character

See also: Character of a representation of a group finished

The characters of the representations lay out, in this context, of a square Produit canonical, it provides a requirement and sufficient convenient to determine the irreducibility of a representation.

* a character is irreducible if and only if its standard by the canonical square product is equal to one.

the demonstration is given in the associated article .

Regular representation

See also: regular Representation

That is to say ( V , ρ) the regular representation of G . It contains all the irreducible representations of G except for an isomorphism, more precisely:

* There exists exactly D subspaces invariants W i of V , of null intersection two to two, such as the restriction of ρ, the regular representation, with W i is isomorphous with ( W , σ).

This decomposition is not single. The number of isomorphous subspaces with W of V is in general higher than D , but it are not all in all direct. There exists nevertheless a single decomposition of the regular representation.

* There exists a single maximum subspace S W of V satisfying all the subspaces isomorph with W . It is called component isotypic W in V .

This decomposition in isotypic components is single for any representation of G , it is called canonical decomposition .

the demonstrations are given in the associated article .

Central function

See also: central Function of a group finished

The concept of central function, i.e. constant function of the group G on each class of conjugation makes it possible to determine the number of irreducible representations exactly:

* There as many exists irreducible representations distinct than from classes of conjugation in the group.

the demonstration is given in the associated article .

Algebra of the group

See also: Algebra of a group finished

The algebra K corresponds to an enrichment of the algebraic structure of the regular representation. The center of the algebra is a ring Commutatif, on which it is possible to use theorems of Arithmétique. They allow, for example to show the following properties:

* the degree of an irreducible representation divides the order of the group.

the demonstration is given in the associated article .

Tensorial product

See also: Produces tensorial and representations of finished groups

The tensorial product introduces a bijection between the representations of two groups G 1 and G 2 and the produces direct G of G 1 and G 2:

* If ( W , σ) is an irreducible representation of G , the group produces direct G 1 and G 2, then there exists an irreducible representation ( W 1, σ1) of G 1 and one ( W 2, σ2) of G 2 such as ( W , σ) is isomorphous with the tensorial product of the two preceding representations. Reciprocally, very produced tensorial of two irreducible representations of G 1 and G 2 is an irreducible representation of G .

the demonstration is given in the associated article .

Induced representation

See also: induced Representation of a group finished

If NR is a normal Sous-groupe normal of G , the induced representations make it possible to establish a relation between ( W , σ) and the restriction of σ NR :

* Either there exists a sub-group H of G container NR and different G such as ( W , σ) is the representation induced by an irreducible representation ( W 1, θ), or the restriction of σ on NR is isotypic.

One from of deduced the following corollary:

* If NR is an abelian Sous-groupe normal of G , then the degree of an irreducible representation divides the order of the Groupe quotient G / NR .

It is moreover to note that the Critère of irreducibility of Mackey provides a requirement and sufficient for an induced representation is irreducible.

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