In Mathematical, and more precisely within the framework of the theory of the representations of a group, the criterion of irreducibility of Mackey proposes a requirement and sufficient so that a induced representation is irreducible Représentation.

This result is named in the honor of the mathematician George Whitelaw Mackey (1916 - 2006) .

Statement

It is necessary to fix the vocabulary and the tools used to state the criterion.

Are G a Groupe finished and H a Sous-groupe of G . K is a commutative body of characteristic either null or first with G the order of the group. If K is of characteristic finished then it algebraic is . In all the cases the Polynomial X ^g - 1 is divided on K .

Either S an element of G and H _s the sub-group intersection of H with the combined of H by S. W is a vector Space on the body K .

$H\_s=sHs^\; \{-\; 1\}\; \backslash \; course\; H$

That is to say ( W , θ) a representation of H and ( V , ρ) the induced representation of G by ( W , θ). θ^s indicates the representation of H _s in the linear Groupe GL ( W ) defined by:

$\backslash \; forall\; H\; \backslash \; in\; H\_s\; \backslash \; quad\; \backslash \; forall\; W\; \backslash \; in\; W\; \backslash \; quad\; \backslash \; theta^s\_h\; (W)\; =\; \backslash \; theta\_\; \{s^\; \{-\; 1\}\; hs\}\; (W)$ One then speaks about combined representation of ( W , θ).

The criterion of Mackey is stated in the following way:

* the representation ( V , ρ) is irreducible if and only if ( W , θ) is irreducible and the various representations θ^s restriction of θ on H _s are disjoined when S is an element of G - H .

There exists a corollary, if the group H is distinguished:

* the induced representation of G by that of H ( W , θ) is irreducible if and only if ( W , θ) is irreducible and isomorphous with no representation θ^s is combined.

Note: This result spreads if the group is topological locally compact and the unit representation in a Espace of Hilbert.

The demonstrations are in the below unrolling box.

Context

Restriction of a representation induced on a sub-group

A natural question is that of the nature of a induced representation on the restriction of ρ on S . For that, the definition of the double classes modulo H and S is necessary:

* a double class of G modulo H and S is a subset E of G such as it exists an element S of G checking E = HS .

It is easy to check that the whole of the double classes form a partition G . One has the same definition as for the classes on the left or on the right:

* a system of representatives C for the double classes is subset of elements C of G such as ScH form a partition of G if C traverses C .

* the restriction on S of the induced representation ( V , ρ) by ( W , θ) is the direct sum of the induced representations of S by ( W , θ^c) if C described C a whole of representatives of the double classes of G modulo H and S .

Reciprocity of Frobenius

See also: Reciprocity of Frobenius

With the notations of the preceding paragraph, the formula of reciprocity of Frobenius is expressed by:

* If ψ and χ indicate the respective characters of θ and ρ:

It is possible to generalize the formula:

* Is F a central function of H and G a central function of G , then the following equality is checked:

Another manner of expressing this property is the following one:

* the application Ind _H^G is the assistant of LMBO _H^G.

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