# Metric Construction of Buildings

Let be a Coxeter group where is generated by elements. Also by for we mean the minimal length is a product decomposition of Let us first use the book definition.

**Definition 1.** A **Weyl distance function** is a map where a set whose elements are called chambers such that

- if and such that then If we also have that then
- if then for any we have a chamber such that and

The pair is called a **– metric space**. The triple is called a **building**.

I’m fairly certain I copied correctly (triple checked), but it looks like or the first sentence in at least. Of importance is the fact we previously mentioned: that chambers of a Coxeter complex coincide with elements of the Coxeter group (since they are (standard) cosets of the trivial standard subgroup). If we thus let then can be thought of as a product on By condition the element is thus not an identity with respect to this product. Also in this regard one can show that the chambers of a Coxeter complex form a building where

with being the gallery metric we previously defined.

Conversely we can say two elements in are **-adjacen**t if and **-equivalent** if they are either -adjacent or equal. If and are -equivalent, we write This is an equivalence relation since

**Proposition 2.** if one takes a generator value.

*Proof.* Suppose By part of the definition there is a such that and Thus by we have and thus

The equivalence classes under are called **-panels**. A **panel** is an -panel for some Galleries can be defined similarly with this terminology. Thus

[1] Abramenko, Peter and Kenneth Brown. *Buildings*. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.

# Coxeter Complexes

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).

[1] Abramenko, Peter and Kenneth Brown. *Buildings*. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.