Modified Ariki-Koike algebra and Yokounuma-Hecke like relations

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji's algebra) $\mathcal H_{n,r}$ over an integral domain $R$ associated with a set of parameters $q,u_1,\ldots,u_r$ in $R$. It turns out that the algebra $\mathcal H_{n,r}$ has a set of generators $t_1,\ldots,t_n$ and $g_1,\ldots g_{n-1}$ subject to a set of defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of $\mathcal H_{n,r}$ which is independent to the choice of $u_1,\ldots u_r$. Hence the algebras associated with the parameters $q, u_1,\ldots u_r$ and $q, u'_1,\ldots u'_r$ are isomorphic even in the case that $(u_1,\ldots u_r)$ and $(u'_1,\ldots u'_r)$ are different. As applications of the presentations, we find an explicit trace form on the algebra $\mathcal H_{n,r}$ which is symmetrising provided the parameters $u_1,\ldots, u_r$ are invertible in $R$. We also show that the symmetric group $\mathfrak S(r)$ acts on the algebra $\mathcal H_{n,r}$, and find a basis and a set of generators of the fixed subalgebra $\mathcal H_{n,r}^{\mathfrak S(r)}$.