# User:Jheald/Irreducible representation/fr

In mathematics,an irreducible representation is a concept used in the context of the theory of Representation of a group.

A 'irreducible representation is a representation that admits only itself and the null representation as sub-representations.

This is important because the Maschke theorem shows that in many cases, representation is direct sum of irreducible representations.

## Definitions and examples

### Definitions

In the rest of the article, G means a group and (V, ρ) a linear representation of G over a field K.

• A representation (V, ρ) is called irreducible if and only if the only invariant subspaces are V and the null vector.
• A character of a representation is called irreducible if and only if the associated representation is.

The representation theory is also expressed in terms of G module. V has naturally a structure of G module in this context, the definition takes the following form:

* A representation (V, ρ) is called irreducible' if and only if V is simple as G module.
* A representation (V, ρ) is called isotype if and only if the only irreducible invariant subspaces other than the null representation are isomorphic pairs.

### Examples

All representations of dimension one are irreducible.

There is only one irreducible and faithful representation of the symmetric group of index three. The articles Representations of the symmetric group of index three and Representations of the symmetric group of index four contain a comprehensive analysis of the irreducible representations of these groups.

If V means a real vector space of dimension two and G the group of linear isometries of V, then the identity of G is an irreducible representation.

## Maschke's theorem

Maschke's theorem states that any irreducible subspace of the representation (V, ρ) is a direct factor, that is to say he has an invariant additional subspace.

This theorem applies in at least two important cases:

In this case,V has the structure of a semi-simple module. Any representation of finite degree of G is then a direct sum of irreducible representations.

The demonstrations are given in the related article.

## Case of a finite group

Assume in this paragraph that G is a finite group g and that the characteristic of K is either zero or coprime with the order of the group. Maschke's theorem then applies. (W, σ) denotes here an irreducible representation of G of degree d. Assume finally that the polynomial Xg - 1 is divided in K.

### Character

(fr:Character of a representation of a finite group)

The characters of the representations have, in this context, a canonical Hermitian product, it provides a necessary and sufficient condition to determine the irreducibility of a representation.

• A character is irreducible if and only if its norm by the canonical Hermitian product is equal to one.

The proof is given in the related article.

### Regular representation

Let (V, ρ) be the regular representation of G. It contains all the irreducible representations of G up to isomorphism, namely:

• There are exactly d invariant subspaces Wi in V, intersection zero pairs, such that the restriction of ρ, the regular representation, to Wi is isomorphic to (W, σ).

This decomposition is not unique. The number of subspaces isomorphic to W for V is generally higher than d, but they are not in direct sum. But there is a unique decomposition of the regular representation.

• There is a unique maximal subspace SW in V containing all the subspaces isomorphic toW. It is called the isotypic component of W in V.

This decomposition into isotypic components is unique for any representation of G; it is called the canonical decomposition.

The demonstrations are given in the related article.

### Class function

(fr:central function of a finite group)

The concept of a class function, that is to say, according to the group G constant on each conjugacy class to determine the exact number of irreducible representations:

• There are as many irreducible representations as distinct conjugacy classes in the group.

The proof is given in the related article.

### Group ring

(fr:algebra of a finite group)

The algebra K[G] corresponds to an enrichment of the algebraic structure of the regular representation. The center algebra is a commutative ring, on which it is possible to use the theorems of arithmetic. They allow, for example, the demonstration of the following properties:

• The degree of an irreducible representation divides order of group.

The proof is given in the related article.

### Tensor Product

(fr:Tensor product and representations of finite groups)

The tensor product introduces a bijection between the representations of two groups G1 and G2 and the direct product G of G1 and G2:

• If (W, σ) is an irreducible representation of G, the group direct product of G1 and G2, then there exists an irreducible representation (W1, σ 1) of G1 and (W2, σ2) of G2 such that (W, σ) is isomorphic to the tensor product of two preceeding representations. Conversely, any tensor product of two irreducible representations of G1 and G2 is an irreducible representation of G.

The proof is given in the related article.

### Induced representation

Where N is a normal subgroup normal G, allow representations induced to establish a relationship between (W, σ) and the restriction σNto:

• Let there exists a subgroup H G to contain N and different as G (W, σ) is the representation induced by an irreducible representation (W1, θ) is the restriction of σ to N is isotypic.

We deduce the following corollary:

It is further noted that the irreducibility criterion Mackey provides a necessary and sufficient condition for an induced representation is irreducible.

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• Either there is a subgroup H G to contain N and different G as (W, σ) is the representation induced by an irreducible representation (W1, θ) is the restriction of σ to N is isotypic.

Let Wi, where i ranges from 1 to n, the canonical decomposition of the restriction of σ to N in isotypic components. We now have the equality:

${\displaystyle W=\bigoplus _{(i=1)}^{n}w_{i}}$

If s is an element of G and if i is an integer between 1 and No, then σ (s)Wi is still an isotypic component. We note that, as Wis an irreducible representation of the G 'action group σ(G) is transitive on the family Wi.

If No is equal to 1, that is to say thatW1 is equal to W, then the restriction of σ to N representation is isotypic.

Otherwise, consider the subgroup H to G consisting of elements leaving invariant globally W1. Let θ 'the restriction of σ to H and the representation of θ H in space W1 equal to θ' toW1. Then (W, σ) is the representation induced by an irreducible representation (W1, θ).

To prove the proposition, a lemma is needed:

* 'If Cis the center groupG, then the degree of an irreducible representation divides the order of quotient group G/C.

Note c the order of the subgroup C to G and (W, σ) an irreducible representation of G. If z is an element of G, then σ (z) commutes with all elements σ(s) where s runs through G. The Schur lemma to conclude that σ(z) is a similarity include λ (z) its report. We note that the application λ is a morphism group C in K *.

Consider now a positive integer m and representation σm tensor product m both the representation σ value inW ${\displaystyle \otimes \cdots \otimes }$ W. If z1, ...,zm is an element of C m then the image of z1 ${\displaystyle \otimes \cdots \otimes }$ zm by σm is a dilatation of ratio λ (z1 ...zm).

Let H sub-group Gm formed elements (z1, ..., zm) as the product of all the coordinates is equal to one. It is a subgroup of the center of Gm, so it is normal. By passing to the quotient, we obtain an irreducible representation of Gm / H. The degree dm of the representation σm is a divider gm /cm-1. This relationship holds for allm, which proves the lemma.

* 'If NOis a normal subgroup of abelian G, then the degree of an irreducible representation divides the order of the quotient group G/N '.

Prove this by induction. The irreducible representation of Gis denoted by (W, σ).

If the restriction of σ toN is isotypic, then asNis abelian and that the only irreducible representation of a finite abelian group is one of degree,image N by σ is composed of dilations. Let G and NO images G and N by σ. Consider the identity representation of G 'valueinW(GL). The preceding lemma shows that the degree of representation that divides the order of the quotient group of G through its center. Now the center has NO because this subgroup is composed of dilations. The level of σ, that is to say, the dimension of W is a divider in the order of G/N. Finally, the canonical mapping of G/N in G/ N is surjective then the order of G /N is a divisor of one of / NG, which completes the proof in this case.

If the restriction of σ to Nis not isotypic, then there exists a group H separate G and N contains, as representation (W,σ) is induced by an irreducible representation (W1, θ) of H. While the degree of representation θ divides the Index [H:N] by induction hypothesis. The degree of σ is equal to θ multiplied by the index [G:H] and thus is a divisor of [G:H].[H:N] and therefore [G:N]. ))

## Notes and references

((Group Representation))