# On the number of constituents of induced modules of Ariki-Koike algebras

## TL;DR

This paper uses categorification and crystal graph analysis to establish new lower bounds on the number of constituents in modules of Ariki-Koike algebras, proving their reducibility.

## Contribution

It introduces a novel approach combining categorification and crystal graphs to determine the irreducible constituents of modules in Ariki-Koike algebras.

## Key findings

- Provides lower bounds on the number of constituents in induced modules.
- Shows that all induced modules are reducible.
- Connects categorification with module decomposition analysis.

## Abstract

We examine the crystal graph of the $\widehat{\mathfrak{sl}}_e$-module arising from an $\widehat{\mathfrak{sl}}_e$-categorification to study the defining endo-functors of the categorification. This yields lower bounds on the number of irreducible constituents of certain objects. We use Ariki's categorification result on Ariki-Koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules. In particular this will imply reducibility of every induced module.