A Note on Scopes Equivalences for Ariki--Koike Algebras as Categorical Actions

TL;DR

This paper explores the relationship between categorical actions of Kac--Moody algebras and Ariki--Koike algebras, focusing on their equivalences and how to translate concepts between these frameworks.

Contribution

It provides a translation framework connecting categorical actions of Kac--Moody algebras with the combinatorics of Ariki--Koike algebras.

Findings

Established a method to translate between categorical actions and Ariki--Koike algebra combinatorics.

Clarified how Morita equivalences relate to categorical actions.

Linked abstract categorical concepts to concrete algebraic structures.

Abstract

A categorical action of a Kac--Moody algebra $\mathfrak{g}$ is built on a category $\mathcal{C}$ decomposed according to the weights $P$ of $\mathfrak{g}$, as well as biadjoint endofunctors $\mathcal{E}_i$ and $\mathcal{F}_i$, abstracting $i$-induction and $i$-restriction, which act on the weight spaces of $\mathcal{C}$ in the same way that the Chevalley generators would act on a regular representation. Chuang and Rouquier initially developed these notions for $\mathfrak{sl}_2$-categorical actions, using them to prove Brou\'e's abelian defect group conjecture for symmetric groups by establishing derived equivalences between blocks of the same defect. In the setting of general categorical actions Webster later showed that many of these derived equivalences are, in fact, $t$-exact, and that, as a result, such an action can be used to separate weight spaces of a categorical action into a finite number of Morita equivalence classes, where these equivalences also preserve decomposition numbers. The combinatorics of these powerful abstract results were concretely established in the case of Ariki--Koike algebras by the first author in arxiv:2301.05153v2, and in this short note we discuss how to translate between the two settings.