Last update: 29 January 2014
In this section we determine the centre of all the Artin semigroups and the centre of the Artin groups of finite type.
7.1. Let be a Coxeter matrix over and the expression of as a disjoint union corresponding to the decomposition of the Coxeter graph into connected components. If is the restriction of to a Coxeter matrix over then (resp. is isomorphic to the direct sum of the (resp. and the centre of each direct sum is isomorphic to the direct sum of the centres of the summands. It suffices therefore to restrict our attention to the case where is irreducible, that is where is connected. In what follows there shall arise the two distinct cases of whether is of finite type, that is is finite, or not.
Let be an irreducible Coxeter matrix. Then we have:
|(i)||If is of infinite type, the centre of the Artin semigroup is trivial.|
|(ii)||If is of finite type, the centre of is an infinite cyclic semigroup. It is generated by the fundamental element if the associated involution is trivial, and otherwise by|
An element in a semigroup (resp. group) with generators shall be called quasicentral when there is, for each a such that The quasicentral elements form a semigroup (resp. group), the quasicentre, in which the centre is naturally embedded.
Now suppose that is a non-trivial element of the quasicentre of and a letter which divides Then for each letter with it is true that for appropriate letters Hence, by 2.1, is divisible by and is therefore divisible by It follows by connectedness of that is divisible by every letter, and hence by 4.1 that there exists a fundamental element and it divides By 5.2, is also quasicentral, and it has strictly smaller length than By induction on the length, there exists a natural number such that This show that the quasicentre of is trivial for the infinite type, and for the finite type is infinite cyclic, generated by Thus we have proven (i) and part of (ii). The rest follows easily from 5.2, as exactly when is central, and is always central since
7.2. From 7.1 we obtain the following description of the centres of the Artin groups of finite type.
Let be an irreducible Coxeter matrix of finite type over Then the centre of the Artin group is infinite cyclic. It is generated by the fundamental element when the associated involution is the identity on and otherwise by
For the generating element of the centre we have resp. where is the Coxeter number and is the product of the generating letters of in any particular order.
Let be quasicentral. As is quasicentral, then is also quasicentral in hence also in Therefore, by 7.1, the element is trivial, and the quasicentre of is infinite cyclic and generated by The rest of the argument follows from 5.2, 5.5 and 5.8.
By explicitly calculating each case under the classification into types and the following may be shown, either by 5.8 or by well-known results about the longest element in (the Coxeter group).
Corollary. In the irreducible case the involution is non-trivial only for the following types: for and
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.