Tits building

t0929001.png ~/encyclopedia/old_files/data/T092/T.0902900 65 0 65 A building is a pair $ ( \Delta ,\ {\mathcal A} ) $ consisting of a simplicial complex $ \Delta $ and a family $ {\mathcal A} $ of subcomplexes $ \Sigma $ ( appartments) satisfying the following conditions: i) each simplex of $ \Delta $ or of any appartment $ \Sigma $ is contained in a maximal simplex (a chamber), and each chamber of $ \Delta $ or $ \Sigma $ has the same finite dimension $ l $ ( the rank of the building); ii) each appartment $ \Sigma $ is connected, in the sense that for any two chambers $ C $ , $ D $ in $ \Sigma $ there is a sequence of chambers of $ \Sigma $ starting with $ C $ and ending with $ D $ , the intersection of any two successive members of which is an $ ( l -1) $ - simplex; iii) any $ ( l -1) $ - simplex of $ \Delta $ ( respectively, of any appartment $ \Sigma $ ) is contained in more than $ 2 $ chambers of $ \Delta $ ( respectively, in exactly $ 2 $ chambers of $ \Sigma $ ); iv) any two chambers $ C $ , $ D $ of $ \Delta $ are contained in some appartment; and v) if two simplices $ A $ , $ B $ of $ \Delta $ are contained in two appartments $ \Sigma $ , $ \Sigma ^ \prime $ , then there is an isomorphism from $ \Sigma $ onto $ \Sigma ^ \prime $ fixing both $ A $ and $ B $ pointwise.

Example 1.

Let $ V $ be a vector space, let $ \Delta $ consist of all chains of non-zero subspaces of $ V $ ( ordered by inclusion), and let each appartment consist of all those chains consisting of subspaces spanned by the non-empty subsets of some basis of $ V $ . Then $ \Delta $ , together with these subcomplexes, is a building.

Example 2.

More generally, let $ G $ be a group with a Tits system $ (G,\ B,\ N,\ S) $ , let $ \Delta $ be the disjoint union of all the cosets $ Pg $ , $ B \leq P<G $ , $ g \in G $ , ordered by reverse inclusion, and let appartments be the sets $ \{ {Pn} : {B \leq P<G, n \in N} \} g $ , $ g \in G $ . Then $ \Delta $ , equipped with these appartments, is a building whose rank is that of the Tits system. Moreover, in this situation $ G $ is transitive on the pairs consisting of an appartment and a chamber of that appartment.

If $ ( \Delta ,\ {\mathcal A} ) $ is a building, then all appartments are isomorphic to the simplicial complex determined by a Coxeter group $ (W,S) $ — the Weyl group of the building, and unique up to isomorphism — as follows: simplices are the right cosets of the subgroups generated by non-empty subsets of $ S $ , where cosets are again ordered by reverse inclusion. In Example 2 this Weyl group is the same as that of the Tits system. A building is called spherical or affine if its Weyl group is. Affine buildings of rank $ 2 $ are just trees in which each vertex is adjacent to at least $ 3 $ others.

Buildings are geometric or combinatorial versions of Tits systems. The most important buildings arise from Tits systems associated with algebraic groups (cf. Algebraic group), although not all buildings do. The two main theorems concerning buildings are classification theorems due to J. Tits [a4], [a5]: 1) any spherical building of rank $ l \geq 3 $ having an indecomposable Weyl group is isomorphic to the building determined by the Tits system of a simple algebraic group; and 2) if $ ( \Delta ,\ {\mathcal A} ) $ is an affine building of rank $ l \geq 4 $ having an indecomposable Weyl group, and if each $ (l -1) $ - simplex is contained in a finite number of chambers, then $ \Delta $ is isomorphic to the complex determined by the affine Tits system of a simple algebraic group over a complete local field. These results stem from the fact that a great deal of geometric information is encoded in the building axioms. In fact, 1) can be viewed as a vast generalization of the classical result that a suitable axiomatization of the notion of projective space leads to a classification in terms of the standard vector space model (cf. Example 1). By contrast, spherical buildings of rank $ 2 $ , and affine buildings of rank $ 3 $ , are too wild to be classifiable: there are many constructions, including free ones. Moreover, spherical buildings of rank $ 2 $ with as Weyl group the symmetric group on $ 3 $ letters correspond naturally to projective planes (incident point-line pairs of a plane corresponding to chambers, cf. also Projective plane).

Buildings are important for the study of the internal structure, representation theory and geometry of simple algebraic groups. They play important roles in the study of finite simple groups and finite geometries as well as in various cohomological questions.

References

[a1]	F. Bruhat, J. Tits, "Groupes réductifs sur un corps local, I. Données radicielles valuées" Publ. Math. IHES , 41 (1972) pp. 5–251 MR0125887
[a2]	W.M. Kantor, "Generalized polygons, SCABs and GABs" L.A. Rosati (ed.) , Buildings and the Geometry of Diagrams (CIME Session, Como 1984) , Lect. notes in math. , 1181 , Springer (1986) pp. 79–158 MR0843390 Zbl 0599.51015
[a3]	M.A. Ronan, "Buildings: main ideas and applications" Bull. London Math. Soc. (To appear) MR1148671 MR1139056 Zbl 0786.51012 Zbl 0753.51009
[a4]	J. Tits, "Buildings of spherical type and finite BN-pairs" , Lect. notes in math. , 286 , Springer (1986) MR0470099 Zbl 0295.20047
[a5]	J. Tits, "Immeubles de type affine" L.A. Rosati (ed.) , Buildings and the Geometry of Diagrams (CIME Session, Como 1984) , Lect. notes in math. , 1181 , Springer (1986) pp. 159–190 MR0843391 Zbl 0611.20026