Parabolic subgroups of Artin groups
Let \((A,\Sigma)\) be an Artin system. For \(X\subseteq\Sigma\) by \(A_X\) is denoted the subgroup of \(A\) generated by \(X\). Such a group is called a parabolic subgroup of \(A\). Van der Lek's theorem is reproved: ``A parabolic subgroup of an Artin group is an Artin group''. An algorithm is given which decides whether two parabolic subgroups of an Artin group are conjugate. Let \(A\) be a finite type Artin group and let \(A_X\) be a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer of \(A_X\) in \(A\) is the set of \(\beta\) in \(A\) such that \(\beta X\beta^{-1}=X\). It is also proved that the commensurator of \(A_X\) in \(A\) is equal to the normalizer of \(A_X\) in \(A\) and that this group is generated by \(A_X\) and the quasi-centralizer of \(A_X\) in \(A\). Moreover, if \(A_X\) is not of type \(D_l\) (\(l\geq 4\) and \(l\) even) then this group is generated by \(A_X\) and the centralizer of \(A_X\) in \(A\).
scientific article; zbMATH DE number 1088589
scientific article