A Property Common to the Weyl and Orthogonal Groups

Abstract

Let W be the Weyl group associated with a root system Σ in an n-dimensional Euclidean space V . In his paper ‘Conjugacy classes in the Weyl group’ [1], Carter shows that every element w ∈ W may be written in the form $$w = {w_{{r_1}}}...{w_{{r_j}}}{w_{{r_{j + 1}}}}...{w_{{r_k}}}$$ (a) where r 1, ..., r k ∈ Σ are linearly independent, and where each of the sets {r 1, ..., r j }, {r j+1, ..., r k } consists of mutually orthogonal roots. The set {r 1, ..., r k } gives rise to a graph, or diagram, similar in form to the Dynkin diagrams; the graphs so arising are useful in describing the conjugacy classes in W , and have certain other significance [2], [3].