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.

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