This week Alice is leading the discussion and the topic is buildings.
We begin with an example. Let and . Also let V be a 4-dimensional vector space with basis . We can form a tetrahedron with vertices labelled by . 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 . 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, is associated with , is associated with and is associated with . 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 we can form the Coxeter complex which is the poset on the set and if and only if . 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 which can be expressed as a union of subcomplexes , called apartments, such that
- (B0) each apartment is a Coxeter complex;
- (B1) for any pair of simplices there exists an apartment containing both;
- (B2) If both contain simplices then there is an isomorphism fixing and pointwise.
Taking and applying (B2) implies that all apartments are isomorphic.
A chamber in the building is a maximal simplex of (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 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 satisfying
- (B0) each apartment is a Coxeter complex
- (B1) for any pair of chambers there exists an apartment containing both
- (B2) If both contain chambers then there is an isomorphism fixing and .
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 we can define a W-distance on the chambers. Given any two chambers , 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 . This is well defined, for if also lie in the apartment A’, by axiom (B2) there is an isomorphism from A to A’ fixing c and d and so .
Going back to the Fano plane example, each apartment is isomorphic to the Coxeter complex for . Letting denote differing by a point and 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 while taking the which goes through the middle we get . Since the W-distances are equal despite taking different apartments.