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

Yanbo Li, Xiangyu Qi

TL;DR

The paper introduces block moving vectors $\mathcal{M}$ as a fine-grained invariant for blocks of Ariki-Koike algebras, encoding abacus-to-core transformations with $w(B)=\sum_i m_i$. It provides a complete classification of representation-finite blocks via the moving-vector data and analyzes derived equivalence phenomena, including blocks with the same weight that lie in different affine Weyl-group orbits. The authors develop an abacus-combinatorics framework—encompassing incomparable abaci and core-compatibility—to study representation types, and apply these ideas to blocks of cyclotomic $q$-Schur algebras, yielding concrete finite-type criteria. Overall, the work extends the understanding of block structure beyond weight alone, linking abacus moves, cores, and Weyl-group actions to representation type and derived categories in cyclotomic settings.

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.

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

TL;DR

The paper introduces block moving vectors as a fine-grained invariant for blocks of Ariki-Koike algebras, encoding abacus-to-core transformations with . It provides a complete classification of representation-finite blocks via the moving-vector data and analyzes derived equivalence phenomena, including blocks with the same weight that lie in different affine Weyl-group orbits. The authors develop an abacus-combinatorics framework—encompassing incomparable abaci and core-compatibility—to study representation types, and apply these ideas to blocks of cyclotomic -Schur algebras, yielding concrete finite-type criteria. Overall, the work extends the understanding of block structure beyond weight alone, linking abacus moves, cores, and Weyl-group actions to representation type and derived categories in cyclotomic settings.

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 -Schur algebras.
Paper Structure (21 sections, 59 theorems, 48 equations, 4 figures)

This paper contains 21 sections, 59 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Poset and representation type
  3. Cellular algebras
  4. Brauer tree algebras
  5. Poset of a finite type self-injective cellular algebra
  6. Abacus orbits and incomparable abaci
  7. Preliminaries on blocks and abaci
  8. Orbits of abaci
  9. Incomparable abaci
  10. Moving vectors
  11. Definition of moving vectors
  12. Simple properties
  13. A construction
  14. Incomparable abaci and moving vectors
  15. Proof of Theorem
  16. ...and 6 more sections

Key Result

Lemma 2.1.3

GL For any $\lambda\in\Lambda$, denote the $A$-module $W_\lambda/\mathop{\mathrm{rad}}\nolimits \lambda$ by $L_{\lambda}$. Let $\Lambda_{0}=\{\lambda\in\Lambda\mid \Phi_{\lambda}\neq 0\}$. Then $\{L_{\lambda}\mid \lambda\in\Lambda_{0}\}$ is a complete set of pairwise non-isomorphic absolutely simple

Theorems & Definitions (123)

  • Definition 2.1.1: GL
  • Definition 2.1.2: GL
  • Lemma 2.1.3
  • Lemma 2.1.4
  • Definition 2.1.5
  • Lemma 2.1.6
  • Lemma 2.1.7
  • proof
  • Lemma 2.2.1
  • Lemma 2.3.1
  • ...and 113 more