# 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.

# Coxeter Complexes

Let $G$ be a Coxeter group with $n$ generators and $J$ be a subset of $\{1,...,n\}.$  We define the subgroup $\langle \{r_i\}_{i\in J}\rangle$ as the standard subgroup $J$ (i.e. hereafter we abuse notation by using $J$ interchangeably for the subset of indicies as well as the subgroup generated by the corresponding elements of $G$).  Its cosets will be called standard cosets of $J$.

Definition 1.  Let $g_1H,g_2J$ be (standard) cosets of standard subgroups $H$ and $J.$  We define the partial ordering $g_1H\leq g_2J\Leftrightarrow g_2J\subseteq g_1H.$  If $G_\Delta$ denotes the set of standard cosets in $G,$ then we call $(G_\Delta,\leq)$ the Coxeter complex of $G,$ and will denote it $G_\Delta$ for short.

It’s clear that $G_\Delta$ is a poset since the ordering is merely reverse containment on all cosets.  Elements of $G_\Delta$ will be called simplices, maximal elements will be called chambers.  Since $1$ is a standard subgroup with $J=\varnothing,$ it follows that the chambers simply coincide with elements of $G.$  Also if $J$ is generated by one generator, then it has two elements: $J=\{1,r_i\}.$  Cosets of such standard subgroups have the form $gJ=\{g,gr_i\},$ and are called panels.  In the case where $g=1,$ we call $1J=J$ the fundamental panel of $J.$ Also if $g$ is a chamber and $J$ is generated by a singleton, then $gJ$ is a face of $g.$  Note that $G$ is the trivial minimal element, but suppose $J$ is generated by $n-1$ generators of $G$, then a coset of $J$ is called a vertex.

Every panel is the face of exactly two chambers:  panels have the form $\{g,gr_i\},$ and are thus faces of $g$ and $gr_i.$

Recall every element in a Coxeter group has the form

$\displaystyle g=\prod_{k=1}^ls_k$

where $s_k=r_i$ for some $i$.  If $g$ is a chamber and we define the chamber $g_i=gs_1\cdots s_i,$ then $g_i$ and $g_{i+1}$ have a panel in common: $\{g_i,g_is_{i+1}=g_{i+1}\}.$  A gallery is a sequence $(g_1,...,g_l)$ of chambers such that $g_i$ and $g_{i+1}$ have a panel in common.  We can thus define a metric on $G$

$\displaystyle d(x,y)=\min\{l:(x=g_1,...,g_l=y)\mbox{~is a gallery}\}.$

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 $\tau:G_V\to\{1,...,n\}$ where $G_V$ denotes the vertices in $G_\Delta$ such that it is a bijection on $G_V(g)=\{gJ\}$ for all $g$ where $G_V(g)$ are the $n$ vertices of the chamber $g.$  The value (or singleton set) $\tau(gJ)$ is called the type of $gJ;$  we may dually call $\{\tau(gJ)\}^C$ the cotype of $gJ.$

The standard type function on a Coxeter complex $G_\Delta$ is defined chamber-wise by

$\tau(gJ)=S-J,$

by which we mean the one element of $S-J,$ as $J$ ranges through $\{1,...,n\}$ (in the sense of which generator it excludes).  Remember we use $J$ both to represent a subset of $\{1,...,n\}$ and to represent the Coxeter subgroup (aka standard subgroup) generated by the elements $\{r_i\}_{i\in J}.$

We can generalize the type function on the Coxeter complex from vertices to all simplices.  We simply map the simplex to the subset of $\{1,...,n\}$ to which all of its vertices are sent.  Hence chambers get sent to the whole set, and have empty cotype.

Definition 3.  Let $gJ$ be a simplex.  We define its link, denoted $lnk(gJ),$ as the set of all simplices $\{g'K\}$ such that $gJ\cap g'K=\varnothing$ and $gJ,g'K$ have a lower bound.

The link is clearly a subcomplex since if $gJ\cap g'K=\varnothing,$ then $gJ\cap S=\varnothing$ for all subsets $S\subseteq g'K.$  Thus the facet ordering is still transitive.

Proposition 4.  Let $gJ$ be a simplex.  Then $lnk(gJ)=J_\Delta$ (as posets).

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