Buildings as Chamber Complexes
4.1 Definition and First Properties
174 4 Buildings as Chamber Complexes
∆ is finite-dimensional, its dimension being the common dimension of its apartments. Note also that ∆ is a chamber complex. For if C and C are maximal simplices, then they are also maximal simplices of some apartmentΣ by (B1), so they have the same dimension and are connected by a gallery.
Any collectionA of subcomplexesΣ satisfying the axioms will be called a system of apartments for ∆. Thus a building is a simplicial complex that admits a system of apartments. Note that we donotrequire that a building be equipped, as part of its structure, with a specific system of apartments. The reason for this is that it turns out that a building always admits a canonical system of apartments. And in the important special case that the apartments are finite Coxeter complexes, it is even true that there is a unique system of apartments. We will prove both of these assertions later in the chapter (Sections 4.5 and 4.7, respectively).
Remark 4.2.The complexes we have called buildings are sometimes called weak buildings in the literature, the term “building” being reserved for the case in which ∆ is thick. This means, by definition, that every panel is a face of at least three chambers. With our definition, by contrast, a building can even be thin. Indeed, a Coxeter complex is a thin building with a single apartment. If we confine ourselves to the thick case, then axiom (B0) can be considerably weakened. Namely, we need only assume that the apartmentsΣ are thin chamber complexes, and it then follows from (B1) and (B2) that they are in fact Coxeter complexes. The proof of this will be given in Section 4.13.
Remark 4.3.Axiom (B2) can be replaced by the following weaker axiom, which is simply the special case of (B2) in which one of the two simplices is a chamber. Some care is needed in the precise formulation, since in the absence of (B2), we do not yet know that all apartments have the same dimension;
thus we need to avoid ambiguity in our use of the word “chamber.”
(B2) Let Σ and Σ be apartments containing simplices A, C, where C is a chamber of Σ. Then there is an isomorphism Σ −→∼ Σ fixing A and C pointwise.
To see that this implies (B2) (in the presence of (B0) and (B1)), consider an arbitrary pair of simplices A, B contained in two apartments Σ and Σ. Choose a chamber C ≥Ain Σ and a chamberD ≥B in Σ, and choose an apartmentΣ containingCand D. Assuming (B2), we have isomorphisms
Σ−→∼ Σ−→∼ Σ,
where the first isomorphism fixes C and B pointwise and the second iso- morphism fixes A and D pointwise. The composite is then an isomorphism Σ−→∼ Σ fixingAandB pointwise, so (B2) holds.
Remark 4.4.Axiom (B2), in turn, is equivalent to the following axiom, which appears at first glance to be stronger:
(B2) Let Σ and Σ be two apartments containing a simplex C that is a chamber of Σ. Then there is an isomorphism Σ −→∼ Σ fixing every simplex in Σ∩Σ.
For suppose that (B2) holds, and letΣ, Σ, andC be as in (B2). Then we have, for eachA∈Σ∩Σ, an isomorphismφA:Σ−→∼ Σ fixingAandC pointwise. But our standard uniqueness argument (Section 3.4.1) shows that there is at most one isomorphism from Σ to Σ fixing C pointwise. So all the φA are equal to a single isomorphism φ, which therefore fixes the entire intersection Σ∩Σ.
Remark 4.5.One can strengthen (B2) still further, by dropping the as- sumption that the two apartments have a common chamber. In other words, the isomorphisms in (B2) can always be taken to fix every simplex in the intersection. We will be able to prove this later in the chapter; see Proposi- tion 4.101 and Exercise 4.108.
Assume, for the remainder of this section, that∆ is a building and that Ais a fixed system of apartments.
Proposition 4.6.∆ is colorable. Moreover, the isomorphisms Σ −→∼ Σ in axiom (B2)can be taken to be type-preserving.
Proof. Fix an arbitrary chamberC, and assign types to its vertices arbitrarily.
If Σ is any apartment containing C, then [since Coxeter complexes are col- orable] the assignment of types onC extends uniquely to a type functionτΣ
ofΣ. For any two such apartmentsΣ, Σ, the type functionsτΣ andτΣ agree on Σ∩Σ; this follows from the fact that τΣ can be constructed asτΣ◦φ, whereφ:Σ −→∼ Σis the isomorphism fixingΣ∩Σ as in (B2). The various τΣ therefore fit together to give a type function τ defined on the union of the apartments containingC. But this union is all of∆ by (B1), so the first assertion of the proposition is proved.
To prove the second assertion, it suffices to consider the isomorphisms that occur in axiom (B2). But such an isomorphism is automatically type-
preserving, since it fixes a chamber pointwise.
Choose a fixed type functionτ on ∆ with values in a setS. In view of the essential uniqueness of type functions, nothing we do will depend in any serious way on this choice. For any apartment Σ, the function τ yields a Coxeter matrix M :=
m(s, t)
s,t∈S, defined by m(s, t) := diam(lkΣA),
where A is any simplex in Σ of cotype {s, t} (see Section 3.2). Since any two apartments are isomorphic in a type-preserving way,M does not depend onΣ:
Proposition 4.7.All apartments have the same Coxeter matrix M.
176 4 Buildings as Chamber Complexes
We will therefore callM theCoxeter matrix of∆. Similarly, we can speak of the Coxeter diagram of ∆; it is a graph with one vertex for eachs ∈ S.
Strictly speaking, we should be talking about the Coxeter matrix and diagram of the pair (∆,A); but we will show in Section 4.4 that the matrix and diagram are really intrinsically associated to ∆ and do not depend on the system of apartmentsA.
The importance of the Coxeter matrix, of course, is that it completely determines the isomorphism type of the apartments. Let’s spell this out in detail: LetWM be the Coxeter group associated toM, with generating setS and relations (st)m(s,t)= 1. In the language of Section 3.5, WM is the Weyl group of every apartment. LetΣM be the Coxeter complexΣ(WM, S). It has a canonical type function with values in S. We can now state the following consequence of Propositions 4.7 and 3.85:
Corollary 4.8.For any apartmentΣ, there is a type-preserving isomorphism Σ∼=ΣM. ThusΣ, endowed with the type functionτ|Σ, is a Coxeter complex of typeM, or of type(WM, S), in the sense of Definition 3.86.
Finally, we record one more simple consequence of the axioms. Recall that the study of local properties of Coxeter complexes consisted of a single result, which said that the link of a simplex in a Coxeter complex is again a Coxeter complex (Proposition 3.16). The situation for buildings is similar:
Proposition 4.9.If ∆ is a building, then so is lkA for any A∈∆. In par- ticular, the link is a chamber complex.
Proof. Choose a fixed system of apartmentsAfor∆. GivenA∈∆, letA be the family of subcomplexes of lk∆Aof the form lkΣA, whereΣis an element of AcontainingA. Any such subcomplex is a Coxeter complex by the result cited above. So it remains to verify (B1) and (B2). Given B, B ∈lk∆A, we can join them with A to obtain simplices A∪B and A∪B in ∆. Since ∆ satisfies (B1), there is an apartment Σ containing both of these simplices.
Hence lkΣA is an element of A containing B and B. This proves that A
satisfies (B1), and the proof of (B2) is similar.
Remark 4.10.The proof, together with the discussion in Section 3.2, tells us how to get the Coxeter diagram of lkA from that of ∆: If A has cotype J ⊆S andD is the Coxeter diagram of∆, then the Coxeter diagram of lkA is the induced diagramDJ with vertex setJ.
As in the case of Coxeter complexes, we can immediately apply the results of Section A.1.4 involving residues:
Corollary 4.11.∆ is completely determined by its underlying chamber sys- tem. More precisely, the simplices of ∆ are in 1–1 correspondence with the residues in C :=C(∆), ordered by reverse inclusion. Here a simplexA corre- sponds to the residueC≥A, consisting of the chambers havingAas a face.
Exercises
4.12.Show that every thin building is a Coxeter complex.
4.13.Let ∆ be a building. For any simplex A ∈ ∆, show that the residue C(∆)≥A is a convex subset ofC(∆) in the sense of Definition 3.92.
4.14. (a) Given a simplex A in a building ∆, one could try to define the support of A by choosing an apartment Σ containing A and declaring suppA to be suppΣA, where the latter is the support of A in Σ (Defi- nition 3.98). Show that this does not work. In other words, if Σand Σ are two apartments containing A, suppΣAneed not equal suppΣA.
(b) Show, on the other hand, that the relation “suppA= suppB” is a well- defined relation on the simplices of ∆, i.e., if A and B have the same support in one apartment containing them, then they have the same support in every apartment containing them.