Coxeter Schemes of Finite Valency and Rank

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. 

The following theorem is the converse of Theorem 6.5.4. It says that Tits systems give rise to Coxeter sets. It makes I lieorem 6.5.4 to be one of our recognition theorems. 

Theorem 12.3.5 A ssume that S is thin and possesses a Tits system (T, J. Then SffT is a Coxeter scheme with respect lo, IffT. 

We are assuming that, for each element j in J, T TjTjT. Thus, for each element j in J, j is an involution in SffT. 

The facts that JffT is constrained and satisfies the exchange condition follow from the hypothesis that TsTjT C TsjTC TsT for any two elements j in J and s in J.  

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In this section, we continue our investigation on Coxeter schemes of finite valency and rank 2 which we started in the previous 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. 

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