Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight module of the quantum enveloping algebra of $\widehat{\mathfrak{sl}_e}$, where $e$ is the multiplicative order of the parameter $q$. Ariki then showed that this isomorphism is actually the identity, if $e\geq 2$ is finite, and Vazirani showed the same for the case $|q|=\infty$.

My question now is the following: Has an analogous result been shown for $q=1$, where we have to replace $e$ by the characteristic of $K$, or rather the additive order of $q=1$?


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