# Decomposition numbers of characteristic 0 Ariki-Koike algebras for q=0 or q generic

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?