Normalisateur et groupe d'Artin de type sphérique

✍ Scribed by Eddy Godelle

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 208 KB
• Volume: 269
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00529-5

✦ Synopsis

Let (A, S) be an Artin group and X a subset of S; denote by A X the subgroup of A generated by X. When A is of spherical type, we prove that the normalizer and the commensurator of A X in A are equal and are the product of A X by the quasi-centralizer of A X in A. Looking the associated monoids A + and A + X , we described the quasi-centralizer of A + X in A + thanks to results in Coxeter groups. These two results generalize earlier results of Paris [J. Algebra 196 (1997) 369-399]. Finally, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin group and in the Coxeter group.