Artin groups and Coxeter groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 26 January 2014

Definition of Artin Groups

In these paragraphs we shall define the Artin groups, and fix some of the notations and ideas which will follow.

1.1. Let I be an index set, FI the free group generated by I and FI+ the free semigroup generated by I inside FI. In the following we drop the subscript I when it is clear from the context.

We call the elements of FI words and the elements of FI+ positive words. The empty word is the identity element of FI+. The positive words have unique representations as products of elements of I and the number of factors is the length L of a positive word. The elements of I are called letters. Frequently we shall denote the letters iI with the more practical notation ai often also with a, b, c, etc. The equivalence relation on positive words A and B is called letterwise agreement and denoted by AB.

In the following we will very often consider positive words with factors beginning with a and in which only letters a and b occur. Such a word of length q will be denoted abq so that abq abaqfactors

1.2. Let M=(mij)i,jI be a Coxeter matrix on I. The Artin group GM corresponding to M is the quotient of FI by the smallest normal subgroup generated by the relations abmab(bamab)-1 where a,bI and mab. In other words: The Artin group GM corresponding to M is the group with generators ai, iI, and the relations aiajmij= ajaimij fori,jI,and mij. When the Coxeter matrix M is clear from the context we drop the index M. We denote the images of the letters and words under the quotient homomorphism FIGM by the same symbols and the equivalence relation on elements A and B in GM is denoted by A=B.

If M is a Coxeter matrix on I then the Coxeter group GM corresponding to M is the group given by generators ai, iI and the relations ai2 = 1 foriI, aiajmij = ajaimij fori,jIwith mij. Obviously this is the same group as the one which is defined by the generators ai, iI, and the usual relations (aiaj)mij =1,fori,jI andmij. The images of the elements A of GM under the canonical homomorphism GMGM are denoted by A and the generating system {ai}iI by I. The pair (GM,I) is a Coxeter System in the sense of Bourbaki [Bou1968] IV 1.3.

In order to describe the Coxeter matrix M we occasionally use the Coxeter graph ΓM in the sense of Bourbaki [Bou1968] IV 1.9.

An Artin group GM is of finite type resp. irreducible resp. of type An, Bn, Cn, Dn etc, when the Coxeter system (GM,I) is finite resp. irreducible resp. of type An, Bn, Cn, Dn etc.

1.3. Let M be a Coxeter matrix. An elementary transformation of positive words is a transformation of the form Aabmab BAbamab B where A,BF+ and a,bI. A positive transformation of length t from a positive word V to a positive word W is a composition of t elementary transformations that begins with V and ends at W. Two words are positive equivalent if there is a positive transformation that takes one into the other. We indicate positive equivalence of V and W by VW.

The semigroup of positive equivalence classes of positive words relative to M is denoted GM+. The quotient homomorphism F+GM+ factors over natural homomorphisms: F+GM+ GM, and for GM of finite type we will show that GM+GM is injective. The equivalence relation on elements V,WGM+ is denoted VW.

Notes and references

This is a translation, with notes, of the paper, Artin-Gruppen und Coxeter-Gruppen, Inventiones math. 17, 245-271, (1972).

Translated by: C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson and A. Ram at the University of Sydney, 1996.

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