Information AboutWeyl Group |
| CATEGORIES ABOUT WEYL GROUP | |
| lie groups | |
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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. Removing the hyperplanes defined by the roots of Φ cuts up 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 chamber, 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}. Weyl groups are examples of Coxeter Group s. This means that they have a special kind of Presentation in which each generator ''xi'' is of order two, and the relations other than ''xi2'' are of the form (''x''''i''''x''''j'')''m''''ij''. The generators are the reflections given by simple roots, and ''mij'' 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 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 : which gives rise to the decomposition of the Flag Variety ''G''/''B'' into Schubert cells (see Grassmannian ). |
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