Coxeter scheme

A scheme which is generated by a Coxeter set L is called a Coxeter scheme with respect to L. 

We shall now see how to obtain Coxeter schemes from thin schemes. 

Theorem 12.3.5 A ssume that S is thin and possesses a Tits system (T, J). Then S//T is a Coxeter scheme with respect to TUT. 

Throughout this section, the letter L stands for a set of involutions. We assume that S is a Coxeter scheme with respect to L, that L = 2, and that S has finite valency. 

From Corollary 7.3.4(H) we also obtain that R is a Coxeter scheme with respect to r (h),rQ (h) satisfying... 

Coxeter schemes is based on the idea of looking at coset geometries of schemes cf. [43 Theorem E]. However, in this monograph, we shall ignore this geometric aspect completely. 

The eleventh chapter is an outline of a general theory of Coxeter schemes. It turns out that most of the basic observations on Coxeter schemes are natural generalizations of the corresponding ones in the theory of Coxeter groups. In fact, the treatment of so-called parabolic subgroups can be taken over almost word-by-word to parabolic subsets. This close relationship between Coxeter schemes and Coxeter groups has a firm mathematical basis, and it leads us to a general question in scheme theory which we had postponed earlier and would like to briefly address here. 

Developing a theory of coset geometries (as mentioned in the previous footnote), one obtains from our recognition theorem of Coxeter schemes Jacques Tits main result on buildings of spherical type . This theorem asserts that each such building is associated with a group if it is thick and of rank at least 3. 

If S consists of two elements and s denotes the non-identity element of S, s is an involution and S is a Coxeter scheme (of rank 1) with respect to . We shall now see how to obtain Coxeter schemes from thin schemes. 

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