In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building.
One can think of this representation as expressing W as some sort of reflection group, with the caveat that B might not be positive definite. It becomes important then to distinguish the representation V from its dual V*. The vectors lie in V, and have corresponding dual vectors in V*, given by: where the angled brackets indicate the natural pairing of a dual vector in V* with a vector of V, and B is the bilinear form as above. Now W acts on V*, and the action satisfies the formula for and any f in V*. This expresses s as a reflection in the hyperplane. One has the fundamental chamber, this has faces the so-called walls,. The other chambers can be obtained from by translation: they are the for. Given a fundamental chamber, the Tits cone is defined to be. This need not be the whole of V*. Of major importance is the fact that the Tits cone X is convex. The action of W on the Tits cone X has fundamental domain the fundamental chamber.
The Coxeter complex
Once one has defined the Tits cone X, the Coxeter complex of W with respect to S can be defined as the quotient of X, with the origin removed, by the positive reals :
The dihedral groups are Coxeter groups, of corresponding type. These have the presentation. The canonical linear representation of is the usual reflection representation of the dihedral group, as acting on a n-gon in the plane. For instance, in the case n = 3, we get the Coxeter group of type, acting on an equilateral triangle in the plane. Each reflection s has an associated hyperplane Hs in the dual vector space, these are the walls. They cut out chambers, as seen below: The Coxeter complex is then the corresponding 2n-gon, as in the image above. This is a simplicial complex of dimension 1, and it can be colored by cotype.
Another motivating example is the infinite dihedral group. This can be seen as the group of symmetries of the real line that preserves the set of points with integer coordinates; it is generated by the reflections in and. This group has the Coxeter presentation. In this case, it is no longer possible to identify V with the dual spaceV*, as B is not positive definite. It is then better to work solely with V*, which is where the hyperplanes are defined. This then gives the following picture: In this case, the Tits cone is not the whole plane, but only the upper half plane. Quotienting out by the positive reals then yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of the infinite dihedral group.
Alternative construction of the Coxeter complex
Another description of the Coxeter complex uses standard cosets of the Coxeter group W. A standard coset is a coset of the form, where for some subset J of S. For instance, and. The Coxeter complex is then the poset of standard cosets, ordered by reverse inclusion. This has a canonical structure of a simplicial complex, as do all posets that satisfy:
