## Artin groups and Coxeter groups

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 $⟨ab⟩q≡ aba⋯⏟q factors$

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 $⟨aiaj⟩mij= ⟨ajai⟩mij fori,j∈I,and mij≠∞.$ 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 $FI⟶GM$ 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 ${\stackrel{‾}{G}}_{M}$ corresponding to $M$ is the group given by generators ${a}_{i},$ $i\in I$ and the relations $ai2 = 1 fori∈I, ⟨aiaj⟩mij = ⟨ajai⟩mij fori,j∈Iwith mij≠∞.$ Obviously this is the same group as the one which is defined by the generators ${a}_{i},$ $i\in I,$ and the usual relations $(aiaj)mij =1,fori,j∈I andmij≠∞.$ The images of the elements $A$ of ${G}_{M}$ under the canonical homomorphism $GM⟶G‾M$ are denoted by $\stackrel{‾}{A}$ and the generating system ${\left\{{\stackrel{‾}{a}}_{i}\right\}}_{i\in I}$ by $\stackrel{‾}{I}\text{.}$ The pair $\left({\stackrel{‾}{G}}_{M},\stackrel{‾}{I}\right)$ 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 $\left({\stackrel{‾}{G}}_{M},\stackrel{‾}{I}\right)$ 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 $A⟨ab⟩mab B⟶A⟨ba⟩mab B$ 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: $F+⟶GM+⟶ GM,$ 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.