Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities ≤ 2, where for nullity 2 there is only one known such presentation. We give a finite presentation for the class of simply laced extended affine Weyl groups. Our presentation is nullity free if rank > 1 and for rank 1 it is given for nullities ≤ 3. The generators and relations are given uniformly for all types, and for a given nullity they can be read from the corresponding finite Cartan matrix and the semilattice involved in the structure of the root system. §0. Introduction Extended affine Weyl groups are the Weyl groups of extended affine root systems which are higher nullity generalizations of affine and finite root systems. In 1985, K. Saito [S] introduced axiomatically the notion of an extended affine root system, he was interested in possible applications to the study of singularities.Extended affine root systems also arise as the root systems of a class of infinite dimensional Lie algebras called extended affine Lie algebras. A systematic study of extended affine Lie algebras and their root systems is given in

We give a finite presentation for reduced non-simply laced extended affine Weyl groups of arbitrary nullity. When nullity is less than or equal to 3, this presentation reduces to a very simple presentation in which the generators and relations can be easily read from a set of data attached to the root system.

Abstract. We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.

There is a well-known presentation for finite and affine Weyl groups called the presentation by conjugation. Recently, it has been proved that this presentation holds for certain sub-classes of extended affine Weyl groups, the Weyl groups of extended affine root systems. In particular, it is shown that if nullity is 2, an A 1 -type extended affine Weyl group has the presentation by conjugation. We set up a general framework for the study of simply laced extended affine Weyl groups. As a result, we obtain certain necessary and sufficient conditions for an A 1 -type extended affine Weyl group of arbitrary nullity to have the presentation by conjugation. This gives an affirmative answer to a conjecture that there are extended affine Weyl groups which are not presented by "presentation by conjugation."

It is known that elliptic Weyl groups, extended affine Weyl groups of nullity 2, have a finite presentation called the generalized Coexter presentation. Similar to the finite and affine case this presentation is obtained by assigning a Dynkin diagram to the root system. Then there is a prescription to read the generators and relations from the diagram. Recently a similar presentation is given for simply laced extended affine Weyl groups of nullity 3 and rank> 1. Employing a new method, we complete this work by giving a similar presentation for nullity 3 extended affine Weyl groups of type A 1 .Keywords Dynkin diagram · Weyl groups · Root system 0 IntroductionIn 1985, K. Saito [10] introduced axiomatically the notion of an extended affine root system and considered the classification of extended affine root systems of nullity 2, which are the root systems equipped with a positive semi-definite quadratic form where the radical of the form has dimension two. Since extended affine root systems of nullity 2 are associated to the elliptic singularities, they are also called elliptic root systems.Extended affine root systems also arise as the root systems of a class of infinite dimensional Lie algebras called extended affine Lie algebras. A systematic study of extended affine Lie algebras and their root systems is given in [1], in particular a set S. Azam ( ) · V. Shahsanaei