Last update: 29 January 2014
In this section we solve the conjugation problem for all Artin groups of finite type.
Two words and are called conjugate when there exists a word such that and we denote this by The conjugation problem consists of giving an algorithm for deciding whether any two given words are conjugate. In our case this problem can easily be reduced to the simpler problem of checking whether any two positive words are conjugate. Here we give a method with which one can calculate, for every positive word the finite set of all positive words which are conjugate to With this the conjugation problem is clearly solved.
8.1. When two positive words and are conjugate, there exists a word such that Since by 5.5 there exist a positive word and a central word such that then also This proves the following lemma.
Positive words and are conjugate precisely when there is a positive word such that
8.2. Every positive word is positive equivalent to the product of square free words. This approaches the goal of creating the positive word conjugate to the positive word in which one conjugates successively by the square-free factors of and in such a manner that one always obtains positive words. By considering 8.1 one arrives at the following construction.
For every finite set of positive words define the set of positive words by Because this set is finite, one can iterate the construction and obtain the sets The sets of positive words are calculable. Since by 5.4, for of finite type there are only finitely many square free words namely divisors of the fundamental word and these are calculable using the division algorithm. Furthermore the division algorithm decides for which square free words the word is left divisible, and the division algorithm 3.6 gives us the quotient Finally, by the solution to the word problem in 6.3, all positive words such that are calculable, that is, all such that Hence is calculable, and so too are
Let be the maximum of the lengths of words in Then it is clear that [Ed: Clearly by putting and for for some ] If we let be the number of positive words of length then has at most elements. Because then eventually, for [Ed: Note that once then for all ] Hence
So is the smallest set of positive words containing both and, for every element all other positive words of the form for some square free word The set is calculable.
Let be a finite set of positive words. Then the finite set is calculable and
By 8.1 it suffices to show, by induction on the length of that for positive words and with for some that is also an element of [Ed: It is clear that One must establish the reverse inclusion. ]
Let where is a square free divisor of of maximal length. We claim that is a left divisor of When we have proved this we are finished because then with and so and by induction hypothesis
To prove the left divisibility of by
For square free, by 5.1 and 5.4 there exists a positive word with Then so that is divisible by We claim that indeed is divisible on the right by every letter and thus by Otherwise by 5.3 we have that is left divisible by and by 3.4 is square free. Both together contradict the maximality of the length of So there exists a positive word with that is with which is what was to be shown.
8.3. The result of the previous section contains the solution to the conjugation problem.
Let be an Artin group of finite type. Let be the fundamental word for Then the following solves the conjugation problem.
|(i)||Let and be arbitrary words. For their exponents take Let and with and positive words. Then and are conjugate when|
|(ii)||If and are positive words, then is conjugate to when is an element of the calculable set of positive words|
The statement (i) follows in a trivial way from the centrality of and (ii) follows trivially from 8.2.
Note added in proof. In the work which was cited in the introduction, Deligne also determined the centre and solved the word and the conjugation problems for the Artin groups of finite type. As we have, he utilises the ideas of Garside but in a geometric formulation which goes back to Tits. We therefore after some consideration deem the publication of our simple purely combinatorial solution defensible.
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.