Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Generalisations of the Tits representation. | Generalisations of the Tits representation Daan Krammer Mathematics Department University of Warwick CV4 7AL Coventry United Kingdom Submitted Aug 21 2007 Accepted Oct 8 2008 Published Oct 20 2008 Mathematics Subject Classification primary 52C35 secondary 20F55 22E40 Abstract We construct a group Kn with properties similar to infinite Coxeter groups. In particular it has a geometric representation featuring hyperplanes simplicial chambers and a Tits cone. The generators of Kn are given by 2-element subsets of 0 . ng. We provide some generalities to deal with groups like these. We give some easy combinatorial results on the finite residues of Kn which are equivalent to certain simplicial real central hyperplane arrangements. 1 Introduction A Coxeter group is a group W presented with generating set S and relations s2 for all s 2 S and at most one relation st m s i for every pair s tg c S where m s t m t s 2 if s t . It is known that then the natural map S W is injective we think of it as an inclusion. We call the pair W S a Coxeter system and S its rank. We generalise this as follows. For any set S let FS denote the free monoid on S. A fully coloured graph is a triple V S m where V S are sets m V X S X S Z 1 u 1g is a map and an action V X FS V written v g vg is specified satisfying the following. o The action of FS on V is transitive. 1 o For all v 2 V s 2 S we have vs s v. o Let v 2 V s t 2 S. Then m v s t 1 if and only if s t. Moreover m v s t m v t s and m v s t m vs s t . Also if k m v s t is finite then v st k v. o The set V is simply 2-connected. That is let V0 S m0 satisfy the above 2 too and let f V0 V be a map satisfying i fv s f vs for all v s ii m0 v s t m fv s t for all v s t. Then f is injective. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R134 1 As the name suggests a graph is involved it has vertex set V and edges x xs of colour s whenever x 2 V s 2 S. In this language 1 means that the graph is connected. Every Coxeter .
