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) \(\mathcal {H}_{n,r}\) ${\mathcal{H}}_{n,r}$ over an integral domain R associated with a set of parameters \(q,u_1,\ldots ,u_r\) $q,u_1,\dots ,u_r$ in R. It turns out that the algebra \(\mathcal {H}_{n,r}\) ${\mathcal{H}}_{n,r}$ has a set of generators \(t_1,\ldots ,t_n\) $t_1,\dots ,t_n$ and \(g_1,\ldots g_{n-1}\) $g_1,\dots g_{n-1}$ subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of \(\mathcal {H}_{n,r}\) ${\mathcal{H}}_{n,r}$ which is independent of the choice of \(u_1,\ldots u_r\) $u_1,\dots u_r$ . As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters \((u_1,\ldots ,u_r)\) $(u_1,\dots ,u_r)$ . We also find an explicit trace form on the algebra \(\mathcal {H}_{n,r}\) ${\mathcal{H}}_{n,r}$ which is symmetrizing provided the parameters \(u_1,\ldots , u_r\) $u_1,\dots ,u_r$ are invertible in R. We show that the symmetric group \(\mathfrak {S}(r)\) $\mathfrak{S}(r)$ acts on the algebra \(\mathcal H_{n,r}\) ${\mathcal{H}}_{n,r}$ , and find a basis and a set of generators of the fixed subalgebra \(\mathcal H_{n,r}^{\mathfrak {S}(r)}\) ${\mathcal{H}}_{n,r}^{\mathfrak{S}(r)}$ .