Weyl groups and reflection geometry

For a finite-dimensional complex semisimple Lie algebra with a chosen Cartan and positive-root system, the Weyl group is generated by reflections in the simple-root hyperplanes. Each reflection fixes its hyperplane pointwise and negates the normal root direction. The coroot pairing supplies the exact coefficient. Reflections preserve the root system, invariant inner product, and weight lattice. Their action on the root span is faithful and finite. The simple reflections generally do not commute; their pairwise products encode angles through Coxeter relations.

In type A₂, realize the root span as triples of real numbers summing to zero, with roots εᵢ−εⱼ. The two simple reflections swap coordinates one and two, or two and three. They generate the full permutation group S₃. The dominant closed chamber consists of nonincreasing triples, so sorting coordinates gives its representative in each orbit. This is a coordinate realization of reflection geometry, not a claim that every Weyl group is a permutation group on exactly three coordinates.

Removing all reflection hyperplanes partitions the real space into open Weyl chambers. The group acts simply transitively on these chambers. Every orbit meets a fixed closed chamber in exactly one point, but a point on its boundary can be fixed by one or more reflections. Its orbit size is the group order divided by its stabilizer order. A regular point lies on no root hyperplane and has trivial stabilizer; a wall point can have a smaller orbit without changing the group or its defining root system.

For integral weights, the same chamber inequalities define dominance, but chamber geometry also applies to nonintegral real points. Integrality is an additional lattice condition needed for finite-dimensional highest-weight classification. Confusing the continuous chamber with the discrete dominant lattice would incorrectly assign a finite-dimensional irreducible representation to every point in a cone. Conversely, a non-dominant weight can still occur inside a finite-dimensional representation; it is the chosen highest weight that must satisfy the classification condition.

Let ρ be half the sum of positive roots. The dot action is defined by w·λ=w(λ+ρ)−ρ. It is an ordinary Weyl action conjugated by translation, not a new reflection group. Its fixed hyperplanes are correspondingly shifted. Highest-weight central characters and the structure of Verma modules naturally involve this action, while weight multiplicities of a finite-dimensional representation are invariant under the ordinary Weyl action. One must say which action appears in a formula rather than relying on the same letter w to disambiguate it.

A Weyl orbit is not generally the full weight set of a representation. For example, an adjoint module contains roots and a zero-weight space, and repeated multiplicities can occur. The orbit of a highest weight often describes its extreme weights, but intermediate weights require additional information. Reflection geometry therefore constrains representation data without replacing character calculations. In higher rank, careful orbit and stabilizer computations are useful checks on those calculations, especially near chamber walls where a generic orbit-count assumption is most likely to fail.