# Metric Construction of Buildings

Let $G$ be a Coxeter group where $G$ is generated by $n$ elements.  Also by $\ell(g)$ for $g\in G$ we mean the minimal length is a product decomposition of $g.$  Let us first use the book definition.

Definition 1.  A Weyl distance function is a map $\delta:C\times C\to G$ where $C$ a set whose elements are called chambers such that

1. $\delta(x,y)=1\Leftrightarrow x=y;$
2. if $\delta(x,y)=g$ and $w\in C$ such that $\delta(w,x)=r_i,$ then $\delta(w,y)\in\{r_ig,g\}.$  If we also have that $\ell(r_ig)=\ell(g)+1,$ then $\delta(w,y)=r_ig;$
3. if $\delta(x,y)=g,$ then for any $i$ we have a chamber $w\in C$ such that $\delta(w,x)=r_i$ and $\delta(w,y)=r_ig.$

The pair $(C,\delta)$ is called a $W$– metric space.  The triple $(C,G,\delta)$ is called a building.

I’m fairly certain I copied correctly (triple checked), but it looks like $3\Rightarrow 2,$ or the first sentence in $2$ 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 $C=C(G_\Delta),$ then $\delta:G\times G\to G$ can be thought of as a product on $G.$  By condition $1,$ the element $1\in G$ is thus not an identity with respect to this product.  Also in this regard one can show that the chambers $C(G_\Delta)$ of a Coxeter complex form a building where

$d(x,y)=\ell(\delta(x,y))$

with $d$ being the gallery metric we previously defined.

Conversely we can say two elements in $C$ are $r_i$-adjacent if $\delta(x,y)=r_i$ and $r_i$-equivalent if they are either $r_i$-adjacent or equal.  If $x$ and $y$ are $r_i$-equivalent, we write $x\sim_{r_i}y.$  This is an equivalence relation since

Proposition 2.  $\delta(x,y)=\delta(y,x)$ if one takes a generator value.

Proof.  Suppose $\delta(x,y)=r_i.$  By part $3$ of the definition there is a $w\in C$ such that $\delta(w,x)=r_i$ and $\delta(w,y)=r_i^2=1.$  Thus by $1$ we have $w=y$ and thus

$\delta(y,x)=\delta(w,x)=r_i=\delta(x,y).$

The equivalence classes under $\sim_{r_i}$ are called $r_i$-panels.  A panel is an $r_i$-panel for some $i.$  Galleries can be defined similarly with this terminology.  Thus $(C,G,\delta)=\left(C(G_\Delta),d\right).$

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