- 期刊
ON THE AFFINE WEYL GROUP OF TYPE A_(n-1)
摘要
We study in this paper the affine Weyl group of type A_(n-1), [1]. Coxeter [1] showed that this group is infinite. We see in Bourbaki [2] that A_(n-1) is a split extension of S_n, the symmetric group of degree n, by a group of translations and of lattice of weights. A_(n-1)is one of the crystallographic Coxeter groups considered by Maxwell [3], [4]. We prove the following: THEOREM 1. A_(n-1), n≥3 is a split extension of S_n by the direct product of (n-1) 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＂_(n-1) coincides with the first A'_(n−1) and so A_(n-1) is not soluble for n＞4. THEOREM 3. The center of A_(n-1) is trivial for n≥3.