In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes 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 group of order 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... A hyperplane is a concept in geometry. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. ...

Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space 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 transitive. 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 {α,γ}. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a symmetry group describes all symmetries of objects. ... See also Simple Lie group. ...

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Weyl groups are examples of Coxeter groups. This means that they have a special kind of 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. In mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ... In mathematics, one method of defining a group is by a presentation. ... See also Simple Lie group. ...

If G is a semisimple linear algebraic group over an algebraically closed field (more generally a split group), and T is a maximal torus, 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 subgroup of G, i.e., a maximal connected solvable subgroup and T is chosen to lie in B, then we obtain the Bruhat decomposition In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, solvable usually refers to the idea of a solvable group, or the corresponding idea of a solvable Lie algebra. ...

which gives rise to the decomposition of the flag variety G/B into Schubert cells (see Grassmannian). In mathematics, a flag is an increasing sequence of subspaces of a vector space. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ...