Group of Weyl

In Mathematical, and in particular in the theory of the algebras of Dregs, the group of Weyl of a System of roots $\ Phi \,$ is the Sous-groupe group of Isométrie S of the system of roots generated by the orthogonal reflections compared to the orthogonal Hyperplan S with the roots.

Example

The system of roots of $A_2 \,$ is consisted of the tops of a regular hexagon centered in the beginning. The complete group of symmetries of this system of roots is consequently the Groupe diédral of order 12. The group of Weyl is generated by the reflections through the lines bissectant the pairs of with dimensions opposed hexagon; it is the diédral group of order 6.

The group of Weyl of a semi-simple Group of Dregs , of a semi-simple Algebra of Dregs , of a linear algebraic Group semi-simple, etc is the group of Weyl of the system of roots of this group or this algebra.

Rooms of Weyl

To remove the hyperplanes defined by the roots of

\ Phi \,

cutting the Euclidean Space in a number finished of opened areas, called the rooms of Weyl . Those are permuted by the action on the group of Weyl, and a theorem establishes that this action is simply transitive. In particular, the number of rooms of Weyl is equal to the order of the group of Weyl. Any vector v different from zero divides Euclidean space into two half spaces bordering the hyperplane

v^ {\ and} \,

orthogonal with v , named

v^ {+} \,

and

v^ {-} \,

. If v belongs to a certain room of Weyl, no root is in

v'^ {\ and} \,

, therefore each root is in

v^ {+} \,

v^ {-} \,

, and if

\ alpha \,

is in one of them, then

- \ alpha \,

is in the other. Thus,

\ Phi^ {+}: = \ Phi \ course v^ {+} \,

made up of exactly half of the roots of

\ Phi \,

. Of course,

\ Phi^ {+} \,

depends on v , but it does not change if v remains in the same room of Weyl.

The bases system of root which respects the choice of $\ Phi \,$ is the simple whole of the roots in $\ Phi^ {+} \,$ , i.e., the roots which cannot be written like a sum of two roots in $\ Phi^ {+} \,$ . Thus, the rooms of Weyl, the unit $\ Phi^ {+} \,$ and base it in determine another, and groups it of Weyl acts simply transitively in each case. The following illustration shows the six rooms of Weyl of a system of roots $A_2 \,$ , a choice of v , the hyperplane $v^ {\ and} \,$ (indicated by a line into dotted) and the positive roots $\ alpha \,$ , $\ beta \,$ , and $\ gamma \,$ . The base in this case is ( $\ alpha \, \ gamma \,$ }.

Groups of Coxeter

The groups of Weyl are examples of the groups of Coxeter. This means that they have a particular kind of presentation in which each generator

x_i \,

is of order two, and the relations other than

x_i^2 \,

are form

(x_i x_j) ^ {m_ {ij}} \,

. The generators are the reflections given by the simple roots and

m_ {ij} \,

is 2,3,4 or 6 dependant if the roots I and J form an angle of 90,120,135 or 150 degrees, i.e., so in the Diagramme of Dynkin, they is not connected, is connected with a simple edge, connected by a double edge or not connected by triple edge. The length of an element of the group of Weyl the length of the shortest word representing this element in terms of these standard generators.

If G is a linear algebraic group semisimple on a Corps algebraically closed (more generally a group deployed ), and T is a maximum Tore, the Normalisateur NR of T contains T like sub-group of finished index and groups it of Weyl W of G is isomorphous with N/T . If B is a Sous-groupe of Borel of G , i.e a related sub-group resolvable maximum selected to contain T , then we obtain a Décomposition of Bruhat

$G = \ bigsqcup_ {W \ in W} BwB \,$

what causes the decomposition of the Variété of flags G / B in cells of Schubert (see Grassmannienne).

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