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 $M$ be a Coxeter matrix over $I$ and $I={\cup}_{\nu}{I}_{\nu}$ the expression of $I$ as a disjoint union corresponding to the decomposition of the Coxeter graph ${\Gamma}_{M}$ into connected components. If ${M}_{\nu}$ is the restriction of $M$ to a Coxeter matrix over ${I}_{\nu},$ then ${G}_{M}^{+}$ (resp. ${G}_{M}\text{)}$ is isomorphic to the direct sum of the ${G}_{{M}_{\nu}}^{+}$ (resp. ${G}_{{M}_{\nu}}\text{),}$ 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 ${\Gamma}_{M}$ is connected. In what follows there shall arise the two distinct cases of whether $M$ is of finite type, that is ${\stackrel{\u203e}{G}}_{M}$ 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 ${G}_{M}^{+}$ is trivial. |
(ii) | If $M$ is of finite type, the centre of ${G}_{M}^{+}$ is an infinite cyclic semigroup. It is generated by the fundamental element $\Delta ,$ if the associated involution $\sigma $ is trivial, and otherwise by ${\Delta}^{2}\text{.}$ |
Proof. | |
An element $Z$ in a semigroup (resp. group) with generators ${a}_{i},$ $i\in I,$ shall be called quasicentral when there is, for each $i\in I,$ a $j\in I$ such that ${a}_{i}Z=Z{a}_{j}\text{.}$ 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 ${G}_{M}^{+},$ and $a$ a letter which divides $Z\text{.}$ Then for each letter $b$ with ${m}_{ab}>2,$ it is true that $baZ\u2a66Zb\prime a\prime \u2a66a(Z:a)b\prime a\prime $ for appropriate letters $a\prime ,b\prime \text{.}$ Hence, by 2.1, $baZ$ is divisible by ${\u27e8ba\u27e9}^{{m}_{ab}},$ and $Z$ is therefore divisible by $b\text{.}$ It follows by connectedness of ${\Gamma}_{M}$ that $Z$ is divisible by every letter, and hence by 4.1 that there exists a fundamental element $\Delta $ and it divides $Z\text{.}$ By 5.2, $(Z:\Delta )$ is also quasicentral, and it has strictly smaller length than $Z\text{.}$ By induction on the length, there exists a natural number $r$ such that $Z\u2a66{\Delta}^{r}\text{.}$ This show that the quasicentre of ${G}_{M}^{+}$ is trivial for the infinite type, and for the finite type is infinite cyclic, generated by $\Delta \text{.}$ Thus we have proven (i) and part of (ii). The rest follows easily from 5.2, as $\sigma =\text{id}$ exactly when $\Delta $ is central, and ${\Delta}^{2}$ is always central since ${\sigma}^{2}=\text{id}\text{.}$ $\square $ |
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 $I\text{.}$ Then the centre of the Artin group ${G}_{M}$ is infinite cyclic. It is generated by the fundamental element $\Delta $ when the associated involution $\sigma $ is the identity on $I,$ and otherwise by ${\Delta}^{2}\text{.}$
For the generating element of the centre we have $\Delta ={\Pi}^{h/2},$ resp. ${\Delta}^{2}={\Pi}^{h},$ where $h$ is the Coxeter number and $\Pi $ is the product of the generating letters of ${G}_{M}$ in any particular order.
Proof. | |
Let $Z={\Delta}^{m\left(z\right)}{Z}^{+}$ be quasicentral. As $\Delta $ is quasicentral, then ${Z}^{+}$ is also quasicentral in ${G}_{M},$ hence also in ${G}_{M}^{+}\text{.}$ Therefore, by 7.1, the element ${Z}^{+}$ is trivial, and the quasicentre of ${G}_{M}$ is infinite cyclic and generated by $\Delta \text{.}$ The rest of the argument follows from 5.2, 5.5 and 5.8. $\square $ |
By explicitly calculating each case under the classification into types ${A}_{n},$ ${B}_{n},$ ${C}_{n},$ ${D}_{n},$ ${E}_{6},$ ${E}_{7},$ ${E}_{8},$ ${F}_{4},$ ${G}_{2},$ ${H}_{3},$ ${H}_{4}$ and ${I}_{2}\left(p\right)$ the following may be shown, either by 5.8 or by well-known results about the longest element in ${\stackrel{\u203e}{G}}_{M}$ (the Coxeter group).
Corollary. In the irreducible case the involution $\sigma $ is non-trivial only for the following types: ${A}_{n}$ for $n\ge 2,$ ${D}_{2k+1},$ ${E}_{6}$ and ${I}_{2}(2q+1)\text{.}$
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.