Modified Ariki-Koike Algebra and Yokonuma-Hecke like Relations

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji’s algebra) H_n,r over an integral domain R associated with a set of parameters q,u₁,…,u_r in R. It turns out that the algebra H_n,r has a set of generators t₁,…,t_n and g₁,…g_n-1 subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of H_n,r which is independent of the choice of u₁,…u_r. As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters (u₁,…,u_r). We also find an explicit trace form on the algebra H_n,r which is symmetrizing provided the parameters u₁,…,u_r are invertible in R. We show that the symmetric group S(r) acts on the algebra H_n,r, and find a basis and a set of generators of the fixed subalgebra H_n,r^S(r).