| Weyl group (mathematics) | |
|---|---|
| Name | Weyl group |
| Caption | Reflection symmetries of a root system |
| Type | Group |
| Field | Mathematics |
| Related | Root system, Coxeter group, Lie algebra, Reflection group |
Weyl group (mathematics)
The Weyl group in mathematics is the finite group generated by reflections associated to a root system; it encodes the symmetry of a Lie algebra or a reductive algebraic group. In the context of Quantum Physics, Weyl groups organize the discrete symmetries of gauge theories, classify degeneracies in quantum spectra, and appear in the structure of representation theory and integrable systems. Their combinatorial and geometric properties make them central to linking algebraic models with physical symmetry operations.
A Weyl group is defined for a reduced root system Φ in a Euclidean space V: it is the group generated by orthogonal reflections s_α across the hyperplanes orthogonal to roots α ∈ Φ. Concretely, s_α(v) = v - 2(α,v)/(α,α) α. The group is finite and acts faithfully on V, preserving the lattice spanned by Φ and the bilinear form. Weyl groups are examples of finite reflection groups and can be presented by generators and relations derived from the angles between simple roots; this leads to a Coxeter presentation in terms of simple reflections. Important basic invariants include the length function, the set of reflections, conjugacy classes, and the invariant polynomial algebra (Chevalley’s theorem relates invariants under the Weyl group to polynomial generators).
For a semisimple Lie algebra g over ℂ with Cartan subalgebra h and root system Φ(g,h), the Weyl group W(g) is obtained as N_G(H)/Z_G(H) for the corresponding Lie group G, or algebraically as generated by reflections s_α for α ∈ Φ. Weyl groups classify Cartan matrices and Dynkin diagrams (types A_n, B_n, C_n, D_n, E6, E7, E8, F4, G2). They control the structure of weight lattices, highest-weight classifications (Weyl character formula), and the linkage principle in Harish-Chandra modules. In quantum settings, Weyl groups determine selection rules for transitions between weight spaces and enter the description of symmetry breaking patterns in spontaneous symmetry breaking and Grand Unified Theory model building.
Weyl groups play a foundational role in the representation theory of reductive groups and quantum groups. The Weyl character formula uses W-summation to compute characters of irreducible highest-weight representations. The group algebra of W deforms to the Hecke algebra H(W,q), a q-parameter family whose representation theory links to Kazhdan–Lusztig theory and intersection cohomology of Schubert varieties. In the theory of quantum groups (Drinfeld–Jimbo algebras) and Yangians, the braid group and R-matrix constructions use Weyl-group combinatorics to produce solutions of the Yang–Baxter equation. In physics, Hecke algebra representations appear in models of anyons, braid statistics, and lattice solvable models such as the XXZ model.
Weyl groups arise in quantum physics through discrete symmetries of Hamiltonians derived from underlying Lie symmetries: they classify degeneracies via root-theoretic selection rules and label multiplets in particle physics. In quantum field theory and gauge theory, Weyl groups act on moduli of vacua and on Cartan subalgebras of gauge groups, affecting monopole and instanton spectra. In solid-state physics, point-group symmetries related to Weyl reflections constrain band crossings; notably the term "Weyl fermion" in condensed matter derives from Weyl's work on chiral fermions but the algebraic Weyl group influences crystal symmetry classifications used in topological semimetals. Weyl group symmetry also appears in the study of exactly solvable models: Bethe ansatz equations, scattering matrices, and S-matrix bootstrap often exploit Weyl reflections and associated Coxeter elements.
Weyl groups are special cases of Coxeter groups with crystallographic restrictions. Their combinatorial structure is encoded by Coxeter matrices and Dynkin graphs. The Bruhat order on W governs the geometry of flag variety Schubert cells and determines singularities of closure relations; these geometric properties are central to computing multiplicities in representation theory and to the geometry of moduli spaces relevant for gauge theories and string compactifications. Coxeter elements, reflection length, and reduced expressions have interpretations in mirror symmetry and in the study of cluster algebras. Actions of W on cohomology rings and equivariant K-theory of G/B (the flag variety) are tools used in both pure mathematics and in localization techniques for path integrals in supersymmetric quantum field theory.
Practical computations with Weyl groups use root datum, Dynkin diagrams, and Coxeter presentations. Standard examples include the symmetric group S_{n+1} as the Weyl group of type A_n, the hyperoctahedral group as type B_n/C_n, and exceptional groups for E8 etc., which appear in string theory and grand-unified model studies. Computational packages in GAP (software), SageMath, and specialized libraries implement Coxeter and Weyl group routines: computing Bruhat order, Kazhdan–Lusztig polynomials, character tables, and orbits on weight lattices. These tools facilitate explicit checks of symmetry constraints in model building, determination of spectral degeneracies, and construction of quantum integrable models using Weyl-reflection symmetries.
Category:Algebraic groups Category:Representation theory Category:Quantum mechanics