Artin groups and Coxeter groups

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

Last update: 29 January 2014

The Centre

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 M be a Coxeter matrix over I and I=νIν the expression of I as a disjoint union corresponding to the decomposition of the Coxeter graph ΓM into connected components. If Mν is the restriction of M to a Coxeter matrix over Iν, then GM+ (resp. GM) is isomorphic to the direct sum of the GMν+ (resp. GMν), 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 M is irreducible, that is where ΓM is connected. In what follows there shall arise the two distinct cases of whether M is of finite type, that is GM is finite, or not.

Let M be an irreducible Coxeter matrix. Then we have:

(i) If M is of infinite type, the centre of the Artin semigroup GM+ is trivial.
(ii) If M is of finite type, the centre of GM+ is an infinite cyclic semigroup. It is generated by the fundamental element Δ, if the associated involution σ is trivial, and otherwise by Δ2.


An element Z in a semigroup (resp. group) with generators ai, iI, shall be called quasicentral when there is, for each iI, a jI such that aiZ=Zaj. The quasicentral elements form a semigroup (resp. group), the quasicentre, in which the centre is naturally embedded.

Now suppose that Z is a non-trivial element of the quasicentre of GM+, and a a letter which divides Z. Then for each letter b with mab>2, it is true that baZZbaa(Z:a)ba for appropriate letters a,b. Hence, by 2.1, baZ is divisible by bamab, and Z is therefore divisible by b. It follows by connectedness of ΓM that Z is divisible by every letter, and hence by 4.1 that there exists a fundamental element Δ and it divides Z. By 5.2, (Z:Δ) is also quasicentral, and it has strictly smaller length than Z. By induction on the length, there exists a natural number r such that ZΔr. This show that the quasicentre of GM+ is trivial for the infinite type, and for the finite type is infinite cyclic, generated by Δ. Thus we have proven (i) and part of (ii). The rest follows easily from 5.2, as σ=id exactly when Δ is central, and Δ2 is always central since σ2=id.

7.2. From 7.1 we obtain the following description of the centres of the Artin groups of finite type.

Let M be an irreducible Coxeter matrix of finite type over I. Then the centre of the Artin group GM is infinite cyclic. It is generated by the fundamental element Δ when the associated involution σ is the identity on I, and otherwise by Δ2.

For the generating element of the centre we have Δ=Πh/2, resp. Δ2=Πh, where h is the Coxeter number and Π is the product of the generating letters of GM in any particular order.


Let Z=Δm(z)Z+ be quasicentral. As Δ is quasicentral, then Z+ is also quasicentral in GM, hence also in GM+. Therefore, by 7.1, the element Z+ is trivial, and the quasicentre of GM is infinite cyclic and generated by Δ. The rest of the argument follows from 5.2, 5.5 and 5.8.

By explicitly calculating each case under the classification into types An, Bn, Cn, Dn, E6, E7, E8, F4, G2, H3, H4 and I2(p) the following may be shown, either by 5.8 or by well-known results about the longest element in GM (the Coxeter group).

Corollary. In the irreducible case the involution σ is non-trivial only for the following types: An for n2, D2k+1, E6 and I2(2q+1).

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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