Let be a Coxeter group with generators and be a subset of We define the subgroup as the standard subgroup (i.e. hereafter we abuse notation by using interchangeably for the subset of indicies as well as the subgroup generated by the corresponding elements of ). Its cosets will be called standard cosets of .
Definition 1. Let be (standard) cosets of standard subgroups and We define the partial ordering If denotes the set of standard cosets in then we call the Coxeter complex of and will denote it for short.
It’s clear that is a poset since the ordering is merely reverse containment on all cosets. Elements of will be called simplices, maximal elements will be called chambers. Since is a standard subgroup with it follows that the chambers simply coincide with elements of Also if is generated by one generator, then it has two elements: Cosets of such standard subgroups have the form and are called panels. In the case where we call the fundamental panel of Also if is a chamber and is generated by a singleton, then is a face of Note that is the trivial minimal element, but suppose is generated by generators of , then a coset of is called a vertex.
Every panel is the face of exactly two chambers: panels have the form and are thus faces of and
Recall every element in a Coxeter group has the form
where for some . If is a chamber and we define the chamber then and have a panel in common: A gallery is a sequence of chambers such that and have a panel in common. We can thus define a metric on
This metric can be extended to any simplices, where it is the minimized version of the above taken over all chambers containing those simplices.
Definition 2. A type function is a map where denotes the vertices in such that it is a bijection on for all where are the vertices of the chamber The value (or singleton set) is called the type of we may dually call the cotype of
The standard type function on a Coxeter complex is defined chamber-wise by
by which we mean the one element of as ranges through (in the sense of which generator it excludes). Remember we use both to represent a subset of and to represent the Coxeter subgroup (aka standard subgroup) generated by the elements
We can generalize the type function on the Coxeter complex from vertices to all simplices. We simply map the simplex to the subset of to which all of its vertices are sent. Hence chambers get sent to the whole set, and have empty cotype.
Definition 3. Let be a simplex. We define its link, denoted as the set of all simplices such that and have a lower bound.
The link is clearly a subcomplex since if then for all subsets Thus the facet ordering is still transitive.
Proposition 4. Let be a simplex. Then (as posets).
 Abramenko, Peter and Kenneth Brown. Buildings. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.