Artin groups and Coxeter groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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, $F_I$ the free group generated by $I$ and $F_I^{+}$ the free semigroup generated by $I$ inside $F_I\text{.}$ In the following we drop the subscript $I$ when it is clear from the context.

We call the elements of $F_I$ words and the elements of $F_I^{+}$ positive words. The empty word is the identity element of $F_I^{+}\text{.}$ 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 $i\in I$ with the more practical notation $a_i$ often also with $a,$ $b,$ $c,$ etc. The equivalence relation on positive words $A$ and $B$ is called letterwise agreement and denoted by $A\equiv B\text{.}$

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 ${⟨ab⟩}^q$ so that $${⟨ab⟩}^q\equiv \underset{\underset{q\hspace{0.17em}\text{factors}}{⏟}}{aba\cdots }$$

1.2. Let $M={\left(m_{ij}\right)}_{i,j\in I}$ be a Coxeter matrix on $I\text{.}$ The Artin group $G_M$ corresponding to $M$ is the quotient of $F_I$ by the smallest normal subgroup generated by the relations ${⟨ab⟩}^{m_{ab}}{\left({⟨ba⟩}^{m_{ab}}\right)}^{-1}$ where $a,b\in I$ and $m_{ab}\ne \infty \text{.}$ In other words: The Artin group $G_M$ corresponding to $M$ is the group with generators $a_i,$ $i\in I,$ and the relations $${⟨a_i{a}_j⟩}^{m_{ij}}={⟨a_j{a}_i⟩}^{m_{ij}}\text{for}i,j\in I,\text{and}{m}_{ij}\ne \infty \text{.}$$ When the Coxeter matrix $M$ is clear from the context we drop the index $M\text{.}$ We denote the images of the letters and words under the quotient homomorphism $$F_I⟶G_M$$ by the same symbols and the equivalence relation on elements $A$ and $B$ in $G_M$ is denoted by $A=B\text{.}$

If $M$ is a Coxeter matrix on $I$ then the Coxeter group ${\bar{G}}_M$ corresponding to $M$ is the group given by generators $a_i,$ $i\in I$ and the relations $$\begin{array}{ccc}{a}_i^2& =& 1\text{for}i\in I,\\ {⟨a_i{a}_j⟩}^{m_{ij}}& =& {⟨a_j{a}_i⟩}^{m_{ij}}\text{for}i,j\in I\text{with}{m}_{ij}\ne \infty \text{.}\end{array}$$ Obviously this is the same group as the one which is defined by the generators $a_i,$ $i\in I,$ and the usual relations $${\left(a_i{a}_j\right)}^{m_{ij}}=1,\text{for}i,j\in I\text{and}{m}_{ij}\ne \infty \text{.}$$ The images of the elements $A$ of $G_M$ under the canonical homomorphism $$G_M⟶{\bar{G}}_M$$ are denoted by $\bar{A}$ and the generating system ${\left\{{\bar{a}}_i\right\}}_{i\in I}$ by $\bar{I}\text{.}$ The pair $({\bar{G}}_M,\bar{I})$ 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 ${\Gamma }_M$ in the sense of Bourbaki [Bou1968] IV 1.9.

An Artin group $G_M$ is of finite type resp. irreducible resp. of type $A_n,$ $B_n,$ $C_n,$ $D_n$ etc, when the Coxeter system $({\bar{G}}_M,\bar{I})$ is finite resp. irreducible resp. of type $A_n,$ $B_n,$ $C_n,$ $D_n$ etc.

1.3. Let $M$ be a Coxeter matrix. An elementary transformation of positive words is a transformation of the form $$A{⟨ab⟩}^{m_{ab}}B⟶A{⟨ba⟩}^{m_{ab}}B$$ where $A,B\in F^{+}$ and $a,b\in I\text{.}$ 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 $W\text{.}$ 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 $V⩦W\text{.}$

The semigroup of positive equivalence classes of positive words relative to $M$ is denoted $G_M^{+}\text{.}$ The quotient homomorphism $F^{+}⟶G_M^{+}$ factors over natural homomorphisms: $$F^{+}⟶G_M^{+}⟶G_M,$$ and for $G_M$ of finite type we will show that $G_M^{+}⟶G_M$ is injective. The equivalence relation on elements $V,W\in G_M^{+}$ is denoted $V⩦W\text{.}$

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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