Moving vectors I: Representation type of blocks of Ariki-Koike algebras

TL;DR

This paper introduces the block moving vector invariant for Ariki-Koike algebra blocks, classifies representation-finite blocks, and explores derived equivalences and representation types of related algebraic structures.

Contribution

It defines a new invariant called block moving vector and uses it to classify blocks, revealing new relationships and distinctions among them.

Findings

Classified representation-finite blocks using block moving vectors.

Identified non-derived equivalent blocks with the same weight and multicharge.

Determined the representation type of blocks in cyclotomic q-Schur algebras.

Abstract

We introduce a new invariant for blocks of Ariki-Koike algebras, called block moving vector, which is a vector of non-negative integers summing up to the weight of the block. In this paper, we use moving vectors to classify representation-finite blocks of Ariki-Koike algebras. As applications, we obtain examples of blocks with the same weight associated with the same multicharge that are not derived equivalent and examples of derived equivalent blocks being in different orbits under the adjoint action of the affine Weyl group. We also determine the representation type for blocks of cyclotomic $q$-Schur algebras.