Spherical Coxeter Sets

This final chapter is the second part of our investigation of Coxeter sets. It deals with spherical Coxeter sets. 

Recall that L is called a Coxeter set if L is constrained and satisfies the exchange condition. Recall also that L is called spherical if S- (L) is not empty. Recall, finally, that a closed subset T of S is called faithfully embedded in S if, for any two elements y in X and z in yT, each faithful map x from y, z to X extends to a bijective map from yT to yyT. 

The first goal of this chapter is to show that (L) is faithfully embedded in S if L is a spherical Coxeter set having at least three elements none of them thin. The corresponding Schur groups turn out to have a Tits system. The situation will be completely described in the corresponding recognition theorem (Theorem 12.3.4). 

In the first section of this chapter, we focus on specific characteristics of spherical Coxeter sets such as maximal elements and conjugation. The second section is devoted to an extension theorem for spherical Coxeter sets. Our approach to this theorem (which follows the line of [46]) is partially inspired by a geometrical reasoning provided by Jacques Tits in [37], 

In the third section of this chapter, we apply results from the two previous sections in order to prove the above-mentioned recognition theorem for spherical Coxeter sets of cardinality at least 3. 

---

In Section 12.4, we shall look closer at the case where L is a spherical Coxeter set consisting of two elements.1 Assuming (L) = S we shall see that... 

In this section, the letter L stands for a spherical Coxeter set. Instead of l we shall write . 

Assuming L to be a spherical Coxeter set we obtain from Lemma 3.6.8 that S-i(L) contains exactly one element. In the following, we shall denote this element by m. ... 

In this section, we look at spherical Coxeter sets containing at least three elements none of them thin. We shall apply Proposition 12.2.7 in order to prove that closed subsets of S generated by such Coxeter sets are faithfully embedded in S. We also establish the corresponding recognition theorem. 

Proposition 12.3.2 Let L be a spherical Coxeter set. Assume that L has at least three elements none of them thin. Let x be an element in X, and let us denote by G the Schur group of (L) with respect to x. Then we have the following. 

Theorem 12.3.4 Let L be a set of involutions of S such that L = S. Assume that L is a spherical Coxeter set consisting of at least three elements none of them thin. Then there exists a finite thin scheme S with a Tits system (T, J) such that S = SffT. 

---