Building (mathematics) - Elementary Properties

Elementary Properties

Every apartment A in a building is a Coxeter complex. In fact, for every two n-simplices intersecting in an (n – 1)-simplex or panel, there is a unique period two simplicial automorphism of A, called a reflection, carrying one n-simplex onto the other and fixing their common points. These reflections generate a Coxeter group W, called the Weyl group of A, and the simplicial complex A corresponds to the standard geometric realization of W. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in A. Since the apartment A is determined up to isomorphism by the building, the same is true of any two simplices in X lie in some common apartment A. When W is finite, the building is said to be spherical. When it is an affine Weyl group, the building is said to be affine or euclidean.

The chamber system is given by the adjacency graph formed by the chambers; each pair of adjacent chambers can in addition be labelled by one of the standard generators of the Coxeter group (see Tits 1981).

Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space. For affine buildings, this metric satisfies the CAT(0) comparison inequality of Alexandrov, known in this setting as the Bruhat-Tits non-positive curvature condition for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths (see Bruhat & Tits 1972).