Weyl group

Weyl group
Name	Weyl group
Caption	The Weyl chambers of the root system of type A<sub>2</sub>, with the associated Weyl group being the dihedral group of order 6.
Field	Lie theory, Group theory, Representation theory
Discovered	Hermann Weyl
Year	1925

Weyl group. In mathematics, particularly in the theory of Lie groups and Lie algebras, a Weyl group is a specific kind of finite group intimately associated with a root system. It is a fundamental Coxeter group generated by reflections corresponding to the simple roots of the system. These groups play a central role in classifying semisimple Lie algebras, understanding their representation theory, and appear in diverse areas including algebraic geometry, invariant theory, and theoretical physics.

Definition and basic properties

A Weyl group is defined for a root system \(\Phi\) within a Euclidean space \(E\). It is the subgroup of the orthogonal group \(O(E)\) generated by reflections through the hyperplanes perpendicular to the roots in \(\Phi\). For a given Cartan subalgebra \(\mathfrak{h}\) of a semisimple Lie algebra \(\mathfrak{g}\), the associated Weyl group can be realized as the quotient group \(N_G(H)/H\), where \(N_G(H)\) is the normalizer of a maximal torus \(H\) in the corresponding Lie group \(G\). Key properties include its finiteness, its status as a Coxeter group with a presentation given by the Cartan matrix, and its faithful action on the weight lattice. The Bruhat order provides a important partial order on its elements.

Examples

The classification of simple Lie algebras over the complex numbers yields a corresponding classification of finite Weyl groups. The primary families are the groups associated with the classical Lie algebras: For type A_n, the Weyl group is the symmetric group \(S_{n+1}\) on \(n+1\) letters. For type B_n and C_n, it is the hyperoctahedral group, isomorphic to the wreath product \(C_2 \wr S_n\). For type D_n, it is a subgroup of the hyperoctahedral group of index two. The exceptional Lie algebras G₂, F₄, E₆, E₇, and E₈ yield the exceptional Weyl groups, with orders 12, 1152, 51840, 2903040, and 696729600, respectively. The dihedral group \(D_6\) is the Weyl group for the root system of type A₂.

Weyl groups in Lie theory

In the structure theory of semisimple Lie algebras, the Weyl group governs the geometry of the root system and the weight lattice. It encodes the symmetries of the Dynkin diagram, which classifies these algebras. The Weyl character formula, a landmark result by Hermann Weyl, expresses the character of an irreducible representation in terms of its highest weight, crucially involving an alternating sum over the group. The Borel–Weil–Bott theorem provides a geometric construction of these representations using the action on flag manifolds. Furthermore, the Bruhat decomposition of a reductive group \(G\) partitions it into cells indexed by elements of the Weyl group.

Weyl groups in Coxeter group theory

Every Weyl group is a finite Coxeter group, but the converse is not true; the dihedral groups \(I_2(p)\) for \(p \neq 2,3,4,6\) are non-crystallographic Coxeter groups. The crystallographic condition, requiring that the Cartan matrix has integer entries, characterizes those Coxeter groups that can arise as Weyl groups. The associated Coxeter–Dynkin diagram is obtained from the Dynkin diagram by omitting arrowheads. The study of their parabolic subgroups, cosets, and the associated Hecke algebra is a major topic. The work of Jacques Tits on buildings provides a profound geometric interpretation of these groups.

Weyl groups in representation theory

Beyond Lie group representations, Weyl groups appear in the representation theory of finite groups of Lie type, such as the general linear group over a finite field. Here, the Deligne–Lusztig theory uses Weyl group elements to construct virtual representations. In the theory of algebraic groups, the Kazhdan–Lusztig polynomials, which have deep connections to intersection cohomology, are indexed by pairs of elements in the Weyl group. The Springer correspondence relates representations of Weyl groups to the geometry of the nilpotent cone in the corresponding Lie algebra.

Weyl groups in geometry and physics

In algebraic geometry, Weyl groups act on the cohomology ring of flag varieties and are central to the study of Schubert calculus. They also appear as monodromy groups in singularity theory. In mathematical physics, particularly in gauge theory and string theory, Weyl groups describe the discrete residual gauge symmetry after spontaneous symmetry breaking in a Yang–Mills theory with a Higgs mechanism. They classify the possible Wilson loop operators and appear in the analysis of D-brane configurations and Calabi–Yau manifold moduli spaces. The McKay correspondence relates finite subgroups of SU(2) to the affine Dynkin diagrams of the extended Weyl groups.

Category:Group theory Category:Lie groups Category:Lie algebras Category:Finite groups