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Blocks of Ariki-Koike algebras and level-rank duality

David Declercq, Nicolas Jacon

TL;DR

The paper addresses the block structure of Ariki-Koike algebras using a generalized notion of $e$-core and $e$-quotient for multipartitions and analyzes the action of the affine Weyl group on blocks via level-rank duality. It develops a comprehensive combinatorial framework based on partitions, symbols, abaci, and Uglov $l$-partitions to parametrize simple modules and blocks, and proves that two $l$-partitions lie in the same block exactly when they share the same $e$-core with the same weight. By relating Chuang–Rouquier derived equivalences to crystal isomorphisms between Fock spaces under level-rank duality, the paper explains block correspondences and classifies the orbits of blocks, showing that different blocks of the same weight need not be derived equivalent when $l>1$. The results yield a complete orbit classification in terms of $l$-multicharges and provide a concrete combinatorial mechanism to study level-rank duality in this modular setting.

Abstract

We study the blocks for Ariki-Koike algebras using a general notion of core for $l$-partitions. We interpret the action of the affine symmetric group on the blocks in the context of level rank duality and study the orbits under this action.

Blocks of Ariki-Koike algebras and level-rank duality

TL;DR

The paper addresses the block structure of Ariki-Koike algebras using a generalized notion of -core and -quotient for multipartitions and analyzes the action of the affine Weyl group on blocks via level-rank duality. It develops a comprehensive combinatorial framework based on partitions, symbols, abaci, and Uglov -partitions to parametrize simple modules and blocks, and proves that two -partitions lie in the same block exactly when they share the same -core with the same weight. By relating Chuang–Rouquier derived equivalences to crystal isomorphisms between Fock spaces under level-rank duality, the paper explains block correspondences and classifies the orbits of blocks, showing that different blocks of the same weight need not be derived equivalent when . The results yield a complete orbit classification in terms of -multicharges and provide a concrete combinatorial mechanism to study level-rank duality in this modular setting.

Abstract

We study the blocks for Ariki-Koike algebras using a general notion of core for -partitions. We interpret the action of the affine symmetric group on the blocks in the context of level rank duality and study the orbits under this action.
Paper Structure (29 sections, 9 theorems, 94 equations)

This paper contains 29 sections, 9 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Partitions, symbols and abaci
  3. Partitions
  4. Abaci and symbols
  5. $l$-Symbol
  6. Symbol of a multipartition
  7. Core and quotient
  8. Nodes
  9. Multipartitions and combinatorial Level-rank duality
  10. A variant of the notion of core and quotient
  11. Relations between the two notions of quotients
  12. Nodes again
  13. Size
  14. Generalized Core
  15. Blocks of Ariki-Koike algebras
  16. ...and 14 more sections

Key Result

Proposition 3.4

Assume that $\lambda$ is a partition and that $\tau^l (\lambda)=({\boldsymbol{\lambda}}^l,{\bf s}^l)$ and $\tau_e (\lambda)=({\boldsymbol{\lambda}}_e,{\bf s}_e)$. Assume that $\mu$ is a partition and that $\tau^l (\mu)=({\boldsymbol{\mu}}^l,{\bf s}^l)$ and $\tau_e (\mu)=({\boldsymbol{\mu}}_e,{\bf s}

Theorems & Definitions (38)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 3.1
  • ...and 28 more