Consider a complex Ariki-Koike algebra $H:=H(q;Q_1,\dots, Q_\ell)$ for some invertible elements $q, Q_1, \dots, Q_\ell$ of $\mathbb{C}$, i.e. an associative $\mathbb{C}$-algebra with generators $T_0, \dots, T_{n-1}$, where
the generators satisfy the braid relations of the Dynkin diagram of type $B_n$, the quadratic relations $(T_i+1)(T_i-q)$ for $1\leq i \leq n-1$, and finally the relation $(T_0-Q_1)\cdots (T_0-Q_\ell)$.
(In particular, $T_1,\dots, T_{n-1}$ generate an Iwahori-Hecke algebra of type $A_{n-1}$ with parameter $q$.)

By the Morita equivalence result of Dipper and Mathas it is sufficient for most questions to only study these algebras in the case of $q$-connected parameters, hence we can assume $Q_i=q^{s_i}$ for some integers $s_i$.

The algebra $H$ is known to be semisimple unless $q$ is not $1$ and a root of unity of order at most $n$ or there exist $i\neq j$ and an integer $-n< d < n$ such that $Q_i q^d=Q_j$.

Now for my questions: If $q$ is a root of unity of order at least $2$ we can compute the decomposition numbers from the Sepcht modules to the simple modules using the LLT algorithm, by Ariki's theorem. What happens if this is not the case, i.e. if $q=1$, or if $q$ is transcendental? Due to the second part of the semisimplicity criterion it might still happen that $H$ is not semisimple. Can we still compute decomposition numbers in thi setting?


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