Tag Archives: coxeter complexes

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.