Weyl chamber

Weyl chamber. In the mathematical fields of Lie theory and representation theory, a Weyl chamber is a fundamental geometric object associated with a root system. It is a maximal connected region in a Euclidean space that is bounded by the hyperplanes orthogonal to the simple roots of the system. The study of these chambers is central to understanding the structure of Lie algebras, their representations, and related Coxeter groups.

Definition and basic properties

A Weyl chamber is formally defined within the context of a root system Φ residing in a Euclidean space E with inner product (·,·). For each root α ∈ Φ, one considers the hyperplane P_α = {λ ∈ E | (λ, α) = 0}. The complement of the union of all these hyperplanes, E \ ∪_{α∈Φ} P_α, decomposes into finitely many connected components. Each of these open, convex, polyhedral cones is called a Weyl chamber. The chambers are in one-to-one correspondence with choices of a set of positive roots or, equivalently, a set of simple roots. The walls of a chamber are precisely the hyperplanes corresponding to the simple roots of the associated positive system. This structure is pivotal in the classification of semisimple Lie algebras via their Dynkin diagrams and is deeply connected to the work of Hermann Weyl and Élie Cartan.

Weyl group action

The Weyl group W, generated by reflections across the hyperplanes P_α, acts simply transitively on the set of Weyl chambers. This means for any two chambers C and C', there exists a unique element w ∈ W such that w(C) = C'. The action of W partitions the ambient space E into a tessellation of chambers, forming a fundamental domain for the action of W on E. The choice of a specific chamber, often called the fundamental Weyl chamber or dominant chamber, corresponds to fixing a set of positive roots. The stabilizer of a point inside a chamber is trivial, while points on walls are stabilized by the Coxeter group generated by the simple reflections corresponding to the walls containing the point. This interplay is a cornerstone in the theory of reflection groups and their associated polytopes like the permutohedron.

Examples in classical Lie algebras

For the Lie algebra 𝔰𝔩(𝑛, ℂ), corresponding to the root system A_{n-1}, the Weyl chambers are cones in a subspace of ℝ^n. They correspond to ordering the coordinates of a vector; for instance, the fundamental chamber may be defined by λ_1 ≥ λ_2 ≥ ... ≥ λ_n with ∑ λ_i = 0. For the Lie algebra 𝔰𝔬(2𝑛, ℂ) (type D_n), chambers are defined by inequalities like ±λ_1 ≥ ±λ_2 ≥ ... ≥ ±λ_n, with specific sign conventions. In the case of 𝔰𝔭(2𝑛, ℂ) (type C_n), the fundamental chamber is given by λ_1 ≥ λ_2 ≥ ... ≥ λ_n ≥ 0. These explicit descriptions are crucial for calculating weights of representations and for the Peter–Weyl theorem in harmonic analysis on compact groups like SU(n) and Spin(n).

Geometric interpretation

Geometrically, a Weyl chamber is a fundamental domain for the action of the Weyl group on the Cartan subalgebra 𝔥 (or its dual space 𝔥*). In the context of a compact Lie group G with maximal torus T, the exponential map identifies the closure of a Weyl chamber in the Lie algebra of T with the set of conjugacy classes in G, a result formalized by the Cartan–Weyl theorem. This provides a geometric picture where the interior of the chamber corresponds to regular conjugacy classes, and its walls correspond to singular ones where the centralizer increases in dimension. This interpretation is vital in symplectic geometry, particularly in the study of coadjoint orbits and their quantization, linking to the Kostant–Kirillov–Souriau theorem.

Applications in representation theory

Weyl chambers are indispensable in the representation theory of Lie algebras. The theorem of the highest weight states that finite-dimensional irreducible representations of a complex semisimple Lie algebra are classified by dominant integral weights, which are precisely the weights lying in the closure of the fundamental Weyl chamber. The Weyl character formula expresses the character of such a representation as an alternating sum over the Weyl group, with terms indexed by weights in the orbit of the highest weight under W. Furthermore, the Kostant multiplicity formula and the study of Verma modules in the BGG category O rely heavily on the chamber geometry. These concepts extend to affine Lie algebras and quantum groups, where the notion of chambers generalizes to the study of crystal bases and Littelmann path models.

Category:Lie theory Category:Representation theory Category:Geometry