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This week Alice is leading the discussion and the topic is buildings.

We begin with an example. Let $W=S_4$ and $S=\{(12),(13),(34)\}$. Also let V be a 4-dimensional vector space with basis $e_1,e_2,e_3,e_4$. We can form a tetrahedron with vertices labelled by $\langle e_1\rangle,\ldots,\langle e_4\rangle$. We place vertices at the midpoint of each edge (these each represent the 2-dimensional subspace spanned by the two 1-spaces at either end of the edge) and a vertex at the midpoint of each face (these represent the 3-spaces spanned by the three 1-spaces at each corner of the face). This gives us a simplicial complex whose 0-simplices are the vertices, the 1-simplices are the lines between 0-simplices, and the 2-simplices are the parts of faces surrounded by the 1-simplices. Now W acts on V by permuting the basis elements. Let I be the 2-simplex surrounded by the vertices $\langle e_1\rangle,\langle e_1,e_2\rangle,\langle e_1,e_2,e_3\rangle$. We can associate each of these three vertices with the subgroup of W generated by the two elements of S which fix the associated subspace. That is, $\langle e_1\rangle$ is associated with $\langle (23),(34)\rangle$, $\langle e_1,e_2\rangle$ is associated with $\langle (12),(34)\rangle$  and $\langle e_1,e_2,e_3\rangle$ is associated with $\langle (12),(23)\rangle$. The 1-simplices joining these vertices are associated with the intersections of the subgroups associated with the end vertices, while we associate I with the identity subgroup. Now for each other simplex we can associate a coset of the appropriate simplex in I.

In general, given a Coxeter system $(W,S)$ we can form the Coxeter complex which is the poset on the set $\{\langle S'\rangle w\mid S'\subseteq S,w\in W\}$ and $B\leqslant A$ if and only if $A\subseteq B$. There was much discussion as to whether or not S needs to be a simple system but this wasn’t resolved.

A building is a simplicial complex $\mathcal{B}$ which can be expressed as a union of subcomplexes $\mathcal{A}$, called apartments, such that

• (B0) each apartment is a Coxeter complex;
• (B1) for any pair of simplices $A,B$ there exists an apartment $\Sigma\in\mathcal{A}$  containing both;
• (B2) If $\Sigma,\Sigma'\in\mathcal{A}$ both contain simplices $A,B$ then there is an isomorphism $\Sigma\rightarrow\Sigma'$ fixing $A$ and $B$ pointwise.

Taking $A,B=\varnothing$ and applying (B2) implies that all apartments are isomorphic.

A chamber in the building is a maximal simplex of $\mathcal{B}$ (where by maximal we do not mean the whole complex).

We say a simplicial complex is thick if every codimension 1 simplex (that is codimension 1 in a chamber) is in at least three chambers.

Theorem If a simplicial complex is thick and contains a union of subcomplexes $\mathcal{A}$ satisfying (B1) and (B2) then (B0) holds, that is, we have a building.

In fact we only need to check that (B1) and (B2) hold for the chambers.

Given a building, we can construct a chamber system (see a previous week’s post)

Equally, we could define a building to be a chamber system which can be expressed as a union of sub-chamber systems $\mathcal{A}$ satisfying

• (B0) each apartment is a Coxeter complex
• (B1) for any pair of chambers $A,B$ there exists an apartment $\Sigma\in\mathcal{A}$  containing both
• (B2) If $\Sigma,\Sigma'\in\mathcal{A}$ both contain chambers $A,B$ then there is an isomorphism $\Sigma\rightarrow\Sigma'$ fixing $A$ and $B$.

A chamber system is called thick if every equivalence class has size at least 3.

The chamber system we get from the Fano plane is drawn below, where each vertex is a chamber and red lines mean the chambers differ by a point, while blue lines mean they differ by a line.

An apartment in this chamber system is shown below.

Given a building whose apartments are isomorphic to the Coxeter complex $(W,S)$ we can define a W-distance on the chambers. Given any two chambers $c,d$, we know from (B1) that there is an apartment A, containing both c and d which correspond to elements  x,y respectively of W. Then $\delta_A(c,d)=yx^{-1}$.  This is well defined, for if $c,d$ also lie in the apartment A’,  by axiom (B2) there is an isomorphism from A to A’ fixing c and d and so $\delta_A(c,d)=\delta_{A'}(c,d)$.

Going back to the Fano plane example, each apartment is isomorphic to the Coxeter complex for $(S_3, \{(12),(23)\})$. Letting $s_1=(12)$ denote differing by a point and $s_2=(23)$ denote differing by a line we can look at the W-distance of two chambers. We take two chambers c,d having no line in common but for which the point of c is the point at the bottom left  corner and the point of d is in the middle of the opposite side of the usual drawing of the Fano plane.

Then using the apartment containing c and d which goes around the bottom  we get $\delta(c,d)=s_1s_2s_1$ while taking the which goes through the middle  we get $\delta(c,d)=s_2s_1s_2$. Since $s_2s_1s_2=s_1s_2s_1$ the W-distances are equal despite taking different apartments.

1. November 19, 2009 8:10 am

I don’t think your Coxeter system (W,S) needs to be simple (I’m reading simple as equivalent to irreducible), and that the only assumption on (W,S) that you want to make is that S is a finite set (at least this is the assumption of Abramenko and Brown’s book “Buildings”).

But it looks like for a reducible Coxeter system, the corresponding complex will just be the product of the complexes for the irreducible components, so you’re not really going to lose anything by restricting to the irreducible case.

November 19, 2009 9:04 am

Hi Peter,

By simple I meant that when look at the corresponding root system, the roots you obtain form a set of simple roots, that is, a minimal spanning set such that all other roots are a linear combination of the simple roots such that the coefficients are either all positive or all negative.

For example, if $W=S_n$ if you take S to the set of all transpositions you are likely to get a different complex to the one you would get it you take S to be the transpositions (12), (23), .. (n-1,n)

A Coxeter system $(W,S)$ is a group W with subset S of involutions and if for each $s_i,s_j\in S$ we let $m_{ij}$ be the order of $s_is_j$ then $W$ is the group defined by the presentation $\langle S\mid (s_is_j)^{m_{ij}}=1\rangle$, where we only use those $m_{ij}$ which are finite. In particular, since $W$ is defined by this presentation, $S$ must be a minimal generating set and if we look at the corresponding root system when represented as a reflection group we do indeed get a simple system of roots.
In the case of symmetric groups which I mentioned earlier, $(S_3, \{(12),(23)\})$ is a Coxeter system for if we let $s_1=(12)$ and $s_2=(23)$ then $S_3$ is defined by the presentation $\langle s_1,s_2\mid s_1^2=s_2^2=(s_1s_2)^3\rangle$. Now if we let $s_3=(13)$ then $S_3=\langle s_1,s_2,s_3\rangle$ and both $s_1s_3$ and $s_2s_3$ have order 3. However, $S_3$ is not defined by the presentation $\langle s_1,s_2,s_3\mid s_1^2=s_2^2=s_3^2=(s_1s_2)^3=(s_1s_3)^3=(s_2s_3)^3\rangle$. This presentation defines an infinite group and indeed, for the elements $s_1,s_2,s_3$ defined in $S_3$ we have the additional relation $s_1s_2s_3=s_2$. (In fact, this presentation defines the affine reflection group $\tilde{A_2}$ which is isomorphic to $\mathbb{Z}^2\rtimes S_3$.) Hence $(S_3, \{s_1,s_2,s_3\})$ is not a Coxeter system.