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Equivalence of $v$-decomposition matrices for blocks of Ariki-Koike algebras

Alice Dell'Arciprete

Abstract

We consider the representation theory of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r \wr \mathfrak{S}_n$. We examine blocks of the Ariki-Koike algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of $n$ whose Specht modules belong to a block $B$ and those of $n-δ_i(B)$ whose Specht modules belong to the block $B'$, obtained from $B$ applying a Scopes' equivalence. This bijection gives us an equivalence for the $v$-decomposition numbers of the Ariki-Koike algebras.

Equivalence of $v$-decomposition matrices for blocks of Ariki-Koike algebras

Abstract

We consider the representation theory of the Ariki-Koike algebra, a -deformation of the group algebra of the complex reflection group . We examine blocks of the Ariki-Koike algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of whose Specht modules belong to a block and those of whose Specht modules belong to the block , obtained from applying a Scopes' equivalence. This bijection gives us an equivalence for the -decomposition numbers of the Ariki-Koike algebras.
Paper Structure (17 sections, 23 theorems, 82 equations)

This paper contains 17 sections, 23 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. The Ariki-Koike algebras
  4. Multipartitions
  5. Kleshchev multipartitions
  6. Specht module and simple modules
  7. Graded decomposition matrices
  8. Blocks of Ariki-Koike algebras
  9. Induction and restriction
  10. $\beta$-numbers and the abacus
  11. Rim $e$-hooks and $e$-core
  12. Background results about weight of multipartitions and core block
  13. Weight and hub of multipartitions
  14. Scopes isometries
  15. Core blocks of Ariki-Koike algebras
  16. ...and 2 more sections

Key Result

Theorem 2.4

djmgl Suppose $\bm\lambda$ and $\bm\mu$ are $r$-multipartitions of $n$ with $\bm\mu$ Kleshchev.

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Example 2.7
  • Theorem 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 38 more