The Ariki--Koike algebras and Rogers--Ramanujan type partitions

In 2000, Ariki and Mathas showed that the simple modules of the Ariki–Koike algebras ${\mathcal{H}}_{\mathbb{C},q;Q_1,\dots ,Q_m}(G(m,1,n))$ (when the parameters are roots of unity and $q\mathrm{\ne }1$) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit this generating function for the $q=-1$ case. This $q=-1$ case is particularly interesting, for the corresponding Kleshchev multipartitions have a very close connection to generalized Rogers–Ramanujan type partitions when $Q_1=\cdots =Q_a=-1$ and $Q_{a+1}=\cdots =Q_m=1$. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m=1$. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block. It is known that these simple modules in a fixed block are labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities when $m=2$.