Weyl group

Weyl group
Name	Weyl group
Type	Finite reflection group
Related	Lie algebra; Coxeter group; root system; Cartan matrix

Weyl group

The Weyl group is a finite reflection group arising in the study of semisimple Lie algebras, algebraic groups, and symmetric structures associated to root systems and Coxeter group theory. It encodes symmetry of a Cartan subalgebra or maximal torus and governs combinatorial and geometric features of flag varietys, representation theory, and algebraic geometry through actions on weight lattices, Bruhat decompositions, and Schubert calculus. The Weyl group links foundational figures and institutions in modern mathematics, appearing in work by Hermann Weyl, Élie Cartan, Élie Joseph Cartan, Claude Chevalley, and groups such as the American Mathematical Society, Institute for Advanced Study, and École Normale Supérieure.

Definition and basic properties

A Weyl group is defined from a root datum associated to a semisimple Lie algebra or a reductive algebraic group via reflections in a real Euclidean space spanned by a Cartan subalgebra. In classical development this construction appears in papers by Hermann Weyl, with subsequent formalization by Élie Cartan and Claude Chevalley. The group is generated by simple reflections corresponding to simple roots and is finite, acting faithfully on the ambient Euclidean space. Important properties were studied at institutions such as Princeton University and University of Göttingen and figure in works connected to the Mathematical Reviews and journals like the Annals of Mathematics.

Root systems and Coxeter presentation

Given a reduced root system attached to a semisimple Lie algebra or a split reductive group over a field, the Weyl group admits a Coxeter presentation with generators s_i subject to relations (s_i s_j)^{m_{ij}} = 1 determined by angles between roots. Historical classification links to contributions from Wilhelm Killing and Élie Cartan and later exposition by Nathan Jacobson and Armand Borel. The Coxeter matrix (m_{ij}) encodes diagrammatic data equivalent to Dynkin diagrams introduced by Eugène Dynkin and used in the classification by Robert Carter and treatments at Harvard University and University of Cambridge.

Weyl groups of Lie algebras and Lie groups

For a complex semisimple Lie algebra g with Cartan subalgebra h, the Weyl group is the quotient of the normalizer of a maximal torus by its centralizer in the corresponding Lie group. This description is central in the study of compact groups like SU(n), SO(n), and Sp(n), as well as exceptional groups such as G2, F4, E6, E7, and E8. The interplay with representation theory appears in work by Harish-Chandra, George Lusztig, and Bertram Kostant, and influences structures studied at the Max Planck Institute for Mathematics and Mathematical Sciences Research Institute.

Coxeter groups, reflections, and length function

Viewed as a finite Coxeter group, the Weyl group is generated by reflections across hyperplanes orthogonal to roots. The length function ℓ(w) relative to simple reflections measures the minimal number of generators needed to express an element and underpins Bruhat order and Hecke algebra constructions studied by Iwahori and Hecke. Fundamental results linking reflection representation, braid relations, and the Tits representation connect to work at University of Paris and centers like the Institut des Hautes Études Scientifiques.

Representations and action on weight lattices

The Weyl group acts naturally on weight and co-weight lattices of a Lie algebra or algebraic group, permuting weights of representations and stabilizing the root lattice. This action is central in the Weyl character formula by Hermann Weyl and its extensions by Harish-Chandra and George Mackey. Interaction with dominant weights, highest weight theory, and tensor product multiplicities appears in work by Robert Steinberg, Igor Frenkel, and Victor Kac, and is applied in contexts at Yale University and University of Chicago.

Bruhat decomposition and Schubert calculus

The Bruhat decomposition of a reductive algebraic group expresses double cosets relative to a Borel subgroup in terms of Weyl group elements; this decomposition organizes flag variety stratifications and Schubert varieties. Schubert calculus computes intersection numbers and cohomology classes indexed by Weyl group elements, with landmark contributions from Schubert and modern developments by William Fulton, Andrei Zelevinsky, and Alexander Grothendieck. Computational and geometric advances appear in collaborations linked to the Clay Mathematics Institute and conferences at Center for Mathematical Sciences.

Examples and classification by type

Classification of Weyl groups follows Dynkin types A_n, B_n, C_n, D_n and exceptional types E6, E7, E8, F4, G2. Classical identifications include symmetric groups S_{n+1} for type A_n, hyperoctahedral groups for B_n and C_n, and even-signed permutation groups for D_n. Exceptional Weyl groups correspond to finite groups tied to exceptional Lie algebras studied by Élie Cartan and in later computational projects at European Mathematical Society and LMS seminars. Applications span connections to modular forms work by Goro Shimura, lattice theory in Conway and Norton research, and string-theoretic appearances in studies by Edward Witten and John Schwarz.

Category:Lie groups Category:Coxeter groups Category:Representation theory