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International Journal of Mathematics and Mathematical Sciences
Volume 10, Issue 1, Pages 147-154
http://dx.doi.org/10.1155/S0161171287000188

On the Affine Weyl group of type A˜n1

Department of Mathematical Sciences, University of Petroleum and Minerals, Dhahran, Saudi Arabia

Received 4 April 1985; Revised 26 March 1986

Copyright © 1987 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study in this paper the affine Weyl group of type A˜n1, [1]. Coxeter [1] showed that this group is infinite. We see in Bourbaki [2] that A˜n1 is a split extension of Sn, the symmetric group of degree n, by a group of translations and of lattice of weights. A˜n1 is one of the crystallographic Coxeter groups considered by Maxwell [3], [4].

We prove the following:

THEOREM 1. A˜n1,  n3 is a split extension of Sn by the direct product of (n1) copies of Z.

THEOREM 2. The group A˜2 is soluble of derived length 3, A˜3 is soluble of derived length 4. For n>4, the second derived group A˜n1 coincides with the first A˜n1 and so A˜n1 is not soluble for n>4.

THEOREM 3. The center of A˜n1 is trivial for n3.