Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups

There are well-known monomorphisms between the Artin groups of finite type An, Bn=Cn and affine type , . The Artin group A(An) is isomorphic to the (n+1)-strand braid group Bn+1, and the other three Artin groups are isomorphic to some subgroups of Bn+1. The inclusions between these subgroups yield monomorphisms A(Bn)→A(An), and . There are another type of monomorphisms A(Bd)→A(Amd−1), A(Bd)→A(Bmd) and A(Bd)→A(Amd) which are induced by isomorphisms between Artin groups of type B and centralizers of periodic braids.In this paper, we show that the monomorphisms A(Bd)→A(Amd−1), A(Bd)→A(Bmd) and A(Bd)→A(Amd) induce injective functions on the set of conjugacy classes, and that none of the monomorphisms A(Bn)→A(An), and does so.