ABSTRACT
A simplicial complex C(W, S) (The Coxeter Complex) is associated with a Coxeter system (W, S), given solely in terms of (W, S), which is naturally endowed with a building structure. This complex admits a canonical geometrical realization in an euclidean space, as a decomposition of this space in simplicial cones, by means of a finite family of hyperplanes. The complex C(W, S) given by the Weyl group W of a system of roots R of a complex semisimple Lie Algebra endowed with the set of generating reflexions S, given by a system of simple roots, is a typical example of a Coxeter System. Its geometrical realization is obtained in the dual space of the real vector space generated by the set of simple roots, by means of the hyperplanes defined by the roots. It is explained how a combinatorial realization is provided by the set A(R) of parabolic subsets of R. As an example remark that the combinatorial realization of the Coxeter Complex C ( S r + 1 , S ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315367309/5389696d-8ee9-4237-ac8b-c1bdba9ca57c/content/eq2402.tif"/> of the general linear group Gl(k r+1) is given by Δ(r) ′ Drap(Ir+1 ). It is recalled that there is a correspondence D↦RD associating to a combinatorial flag D a parabolic set of roots in R(Ir+1 ) = Ir+1 × Ir+1 /Δ(Ir+1 ). The interest of the combinatorial realization is that it is directly connected with the Tits geometry associated with the building, and appears as a Galois geometry of the characteristic equation of a generic element of the Lie algebra acting by the adjoint action (cf. [4], Note historique).
