Weyl group: Facts, Discussion Forum, and Encyclopedia Article

Weyl group

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	Topics Weyl group Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , in particular the theory of Lie algebraLie algebra Summary In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... s, the Weyl group of a root systemRoot system Overview In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.... Φ is a subgroupSubgroup In group theory, given a group G under a binary operation , we say that some subset H of G is a subgroup* of G... of the isometryIsometry In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism be... group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplaneHyperplane A hyperplane is a concept in geometry.... s orthogonal to the roots. For example, the root system of A₂ consists of the vertices of a regular hexagon centered at the origin. The full group of symmetries of this root system is therefore the dihedral groupDihedral group In mathematics, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group i... of orderOrder (group theory) In group theory, a branch of mathematics, the term *order* is used in two closely related senses:... 12. The Weyl group is generated by reflections through the lines bisecting pairs of opposite sides of the hexagon; it is the dihedral group of order 6. The Weyl group of a semi-simple Lie groupLie group In mathematics, a Lie group is a continuous group, in the sense that the group elements have the topology of a manifold, an... , a semi-simple Lie algebraLie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... , a semi-simple linear algebraic groupLinear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is define... , etc. is the Weyl group of the root system of that group or algebra. Removing the hyperplanes defined by the roots of Φ cuts up Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... into a finite number of open regions, called Weyl chambers. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitiveGroup action In mathematics, a symmetry group describes all symmetries of objects.... . In particular, the number of Weyl chambers equals the order of the Weyl group. Any non-zero vector v divides the Euclidean space into two half-spaces bounding the hyperplane v^∧ orthogonal to v, namely v⁺ and v⁻. If v belongs to some Weyl chamber, no root lies in v^∧, so every root lies in v⁺ or v⁻, and if α lies in one then −α lies in the other. Thus Φ⁺ := Φ∩v⁺ consists of exactly half of the roots of Φ. Of course, Φ⁺ depends on v, but it does not change if v stays in the same Weyl chamber. The base of the root system with respect to the choice Φ⁺ is the set of simple roots in Φ⁺, i.e., roots which cannot be written as a sum of two roots in Φ⁺. Thus, the Weyl chambers, the set Φ⁺, and the base determine one another, and the Weyl group acts simply transitively in each case. The following illustration shows the six Weyl chambers of the root system A₂, a choice of v, the hyperplane v^∧ (indicated by a dotted line), and positive roots α, β, and γ. The base in this case is . Weyl groups are examples of Coxeter groupCoxeter group In mathematics, a Coxeter group, named after H.S.M.... s. This means that they have a special kind of presentationPresentation of a group In mathematics, one method of defining a group is by a presentation.... in which each generator x_i is of order two, and the relations other than x_i² are of the form (x_ix_j)^m_ij. The generators are the reflections given by simple roots, and m_ij is 2, 3, 4, or 6 depending on whether roots i and j make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the Dynkin diagram they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge. The length of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. If G is a semisimple linear algebraic group over an algebraically closed fieldAlgebraically closed field In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with co... (more generally a split group), and T is a maximal torusMaximal torus In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups.... , the normalizer N of T contains T as a subgroup of finite index, and the Weyl group W of G is isomorphic to N/T. If B is a Borel subgroupBorel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected so... of G, i.e., a maximal connectedConnected space In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the dis... solvableSolvable In mathematics, solvable usually refers to the idea of a solvable group, or the corresponding idea of a solvable Lie algebra... subgroup and T is chosen to lie in B, then we obtain the Bruhat decompositionBruhat decomposition In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gaus... which gives rise to the decomposition of the flag variety G/B into Schubert cells (see GrassmannianGrassmannian In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, o... ).