Artin groups and Coxeter groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $I={\cup }_{\nu }{I}_{\nu }$ 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 ${\bar{G}}_M$ 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 $baZ⩦Zb\prime a\prime ⩦a(Z:a)b\prime a\prime$ for appropriate letters $a\prime ,b\prime \text{.}$ Hence, by 2.1, $baZ$ is divisible by ${⟨ba⟩}^{m_{ab}},$ 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 $Z⩦{\Delta }^r\text{.}$ 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 ${\bar{G}}_M$ (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 $I_2(2q+1)\text{.}$

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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