Coxeter Sets

In the last two sections of this chapter, we impose specific conditions on sets of involutions. In Section 3.5, we look at constrained sets of involutions, and in the last of the six sections of this chapter, we look at constrained sets of involutions which satisfy the exchange condition. According to what we said in the preface of this monograph, such sets of involutions are called Coxeter sets. 

It might be worth mentioning that, via the group correspondence, most of the results about Coxeter sets which we compile in the last section are natural generalizations of well-known facts on Coxeter groups. 

We call L a Coxeter set if L is constrained and satisfies the exchange condition. 

We consider Theorem 3.6.4 and Theorem 3.6.6 to be the main results of this section. The equations given in these theorems are crucial for our approach to Coxeter sets. 

We are assuming that L is a Coxeter set. In particular, L is constrained. Under this hypothesis, one can modify the exchange condition, and we shall do this in the following lemma. 

In the last section of this chapter, in Section 8.7, we present identities about roots of unity in integral domains. The results will be useful in Section 12.4 where we shall investigate Coxeter sets of cardinality 2. 

Later in this chapter, in Section 10.6, we shall assume additionally that (L) has finite valency. It is one of the main goals of this chapter to show that (in this case) L is a Coxeter set (defined in Section 3.6) or a Moore set. (The definition of a Moore set will be given in Section 10.6.) This goal will be achieved in Theorem 10.6.6. 

A general theory of Coxeter sets will be developed in the last two chapters of this monograph. A more detailed investigation of the first case of Theorem 10.6.6 will be part of this more general approach. 

The results in this section show that the notion of a Coxeter set emerges naturally from the theory of dihedral closed subsets. On this occasion, it might be worthwhile to recall that the theory of Coxeter sets is equivalent to the theory of buildings in the sense of Jacques Tits. 

Theorem 10.6.6 (together with Corollary 10.5.2(h)) says that Coxeter sets as well as Moore sets are characterized by the equation 5 i(L) = 1. 

The following corollary is a representation theoretic characterization of Coxeter sets and Moore sets. It was proved first in [42 Theorem B]. Here, we obtain it as a corollary of Theorem 10.6.6 and Theorem 10.6.7. 

Corollary 10.6.8 A ssume that there exists an algebraically closed field C of characteristic 0 with C[L] = C(L). Then L is a Coxeter set or a Moore set. 

In this chapter, we start to look more thoroughly at Coxeter sets. In order to do so we first recall the definition of a Coxeter set. 

The goal of the last of the four sections of this chapter is the proof of a specific extension theorem (Theorem 11.4.6) for Coxeter sets. 

The following lemma generalizes Lemma 3.4.8 for Coxeter sets. 

Theorem 11.2.4 tells us that, in order to investigate Coxeter sets without thin elements, it is enough to look at Coxeter sets which generate a simple closed subset. 

The following lemma is a result about simple closed subsets generated by a Coxeter set. It will be needed in the proof of Lemma 12.3.1. 

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. ... 

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