Weyl GroupEdit

The Weyl group is a central object in the theory of semisimple Lie algebras and their associated geometric and algebraic structures. It is the finite group generated by reflections corresponding to the roots of a root system, and it sits at the crossroads of geometry, combinatorics, and representation theory. The Weyl group is a concrete way to understand the symmetries of a Lie algebra’s structure, and it also appears as a key example of a finite Coxeter group.

The concept is named after Hermann Weyl, who introduced these reflection symmetries in the context of classifying and studying Lie groups and their representations. In modern language, the Weyl group acts on a real Euclidean space that carries a root system, preserving the system and generating its full symmetry. This action can be realized explicitly as a group generated by reflections sα for each root α, where sα(v) = v − ⟨v, α∨⟩ α and α∨ = 2α / ⟨α, α⟩ is the coroot.

Definition and origins

The Weyl group W is the group generated by reflections corresponding to the roots of a root system Φ in a Euclidean space V with inner product ⟨·,·⟩. The reflection sα in Φ fixes the hyperplane orthogonal to α and maps α to −α.
The root system Φ is a finite set of nonzero vectors in V that is closed under reflections sα and scaled so that Φ ∩ Rα = {±α} in a way that encodes the angles between roots.
Each Lie algebra of semisimple type has an associated root system, with a finite set of simple roots {α1, …, αr} that form a basis for the space spanned by Φ. The Weyl group is generated by the simple reflections si := sαi.
The arrangement of simple roots is summarized by a Dynkin diagram, which encodes the angles between simple roots and thus determines the relations among the generators si. The classification of these diagrams yields the familiar families A, B, C, D and the exceptional types E, F, G.

For those who think in terms of more combinatorial pictures, the Weyl group is a finite Coxeter group, presented by generators si with relations (si)^2 = 1 and (si sj)^{mij} = 1 for i ≠ j, where the integers mij are determined by the angle between αi and αj. The Dynkin diagram records these exponent relations, and the geometry of the group action is reflected in the arrangement of reflecting hyperplanes.

Root systems, Dynkin diagrams, and types

A root system Φ lives in a real Euclidean space V and satisfies reflection symmetry with respect to all its roots. The Weyl group W is the symmetry group generated by the reflections sα for α in Φ.
The simple roots α1, …, αr produce a root lattice QR = Zα1 ⊕ … ⊕ Zαr and a weight lattice P that sits between QR and its dual lattice. The Weyl group preserves both lattices and acts by automorphisms.
Dynkin diagrams encode the angular relationships among simple roots and thus the defining relations of the Weyl group. The classical irreducible types are:
- An (n ≥ 1): W ≅ Sn+1, the symmetric group on n+1 letters.
- Bn and Cn (n ≥ 2): W is the hyperoctahedral group, the group of signed permutations, of order 2^n n!.
- Dn (n ≥ 4): W is the subgroup of even signed permutations, of order 2^{n-1} n!.
- Exceptional types: E6, E7, E8, F4, G2, with corresponding finite groups of large but explicit orders.
In each case, the Weyl group acts as a finite reflection group preserving the root system and the lattices, and it captures the combinatorial structure of the root data.

For readers who want to connect the algebraic data to geometry, the root system and its Weyl group give a concrete realization of the symmetries of the corresponding Lie algebra. See for example root system and Dynkin diagram for foundational pictures, and consider the type classifications A_n, B_n, C_n, D_n, E6, E7, E8, F4, G2 for concrete instances.

Geometry: chambers, alcoves, and actions

The action of the Weyl group partitions the ambient space V into regions called Weyl chambers. Each chamber is a connected component of the complement of the reflection hyperplanes. The fundamental chamber is the region where all simple roots take nonnegative pairing with vectors in V.
The Weyl group acts simply transitively on the set of chambers: any chamber can be obtained from the fundamental chamber by a unique element of W.
More refined geometric pictures involve alcoves in the affine setting, where the affine Weyl group acts by reflections in an extended arrangement of hyperplanes. This viewpoint is crucial in several areas, including representation theory and combinatorics.
The action preserves the root and weight lattices and encodes the symmetry of the associated Lie algebra’s weight data. See Weyl chamber and alcove for geometric pictures, and weight lattice for lattice-theoretic language.

Invariants, representations, and structure

The Weyl group acts on multiple natural spaces: the root lattice QR, the weight lattice P, and the polynomial ring S(V*) of polynomial functions on V. Invariant theory studies the polynomials fixed by W, leading to rich algebraic structure.
Chevalley’s theorem (in its various formulations) describes the invariant ring S(V*)^W as a polynomial algebra on r generators, where r is the rank of the root system. The degrees of these basic invariants are closely tied to the exponents of the root system.
The representation theory of semisimple Lie algebras is organized by the action of the Weyl group on weight spaces. The Weyl group permutes weights, and its action helps determine characters, multiplicities, and branching rules for representations.
The Weyl group also interacts with geometric and combinatorial objects such as the Bruhat order, which encodes a partial order on the elements of W via reduced expressions in simple reflections. See Bruhat order for more on this combinatorial structure.

Connections and generalizations

Affine Weyl groups generalize the finite Weyl groups by incorporating translations along the root lattice, leading to infinite Coxeter groups with rich structure used in representation theory, number theory, and mathematical physics. See Affine Weyl group.
The Weyl group sits inside the larger landscape of finite Coxeter groups, which classify reflection symmetries in Euclidean spaces. See Coxeter group for a broader perspective.
In the broader Lie theory context, Weyl groups relate to Kac–Moody algebras, quantum groups, and various geometric constructions such as flag varieties and Schubert calculus. See Kac–Moody algebra and Flag variety for connections.

Examples and explicit descriptions

Type A_n: W ≅ S_{n+1}, realized as permutation symmetry on the coordinates of an (n+1)-dimensional space modulo the diagonal scope, corresponding to the root system of type A_n.
Type B_n/C_n: W is the group of all signed permutations of n coordinates, of order 2^n n!, reflecting the duality between these two non-simply-laced families.
Type D_n: W is the subgroup of signed permutations with an even number of sign changes, of order 2^{n-1} n!.
Exceptional types E6, E7, E8, F4, G2: finite groups with well-known, highly symmetric root systems and rich combinatorial structure; their Weyl groups appear in various mathematical and physical contexts.