On the Affine Weyl group of typeA˜n−1

Abstract

We study in this paper the affine Weyl group of typeA˜n−1, [1]. Coxeter [1] showed that this group is infinite. We see in Bourbaki [2] thatA˜n−1is a split extension ofSn, the symmetric group of degreen, by a group of translations and of lattice of weights.A˜n−1is one of the crystallographic Coxeter groups considered by Maxwell [3], [4].We prove the following:THEOREM 1.A˜n−1,  n≥3is a split extension ofSnby the direct product of(n−1)copies ofZ.THEOREM 2. The groupA˜2is soluble of derived length3,A˜3is soluble of derived length4. Forn>4, the second derived groupA˜″n−1coincides with the firstA˜′n−1and soA˜n−1is not soluble forn>4.THEOREM 3. The center ofA˜n−1is trivial forn≥3.