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.