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Decomposition numbers for weight 3 blocks of Iwahori--Hecke algebras of type B

Matthew Fayers, Lorenzo Putignano

TL;DR

The paper proves that in a weight $3$ block $B$ of an Iwahori--Hecke algebra of type $B$, all decomposition numbers are in {0,1} over any field. The authors develop a detailed combinatorial framework based on bipartitions, the abacus, and restricted bipartitions, and they leverage Brundan--Kleshchev branching rules together with the cyclotomic Jantzen--Schaper formula to bound multiplicities. A careful block-by-block analysis, distinguishing core vs non-core blocks and four block types (II–IV), reduces the problem to a finite number of exceptional bipartitions whose contributions are controlled via JS-order arguments and duality. The results yield a complete, inductive proof that extends known type A weight-$3$ results to type $B$ and informs efficient computation of decomposition numbers via the Jantzen--Schaper formula, with implications for Ariki--Koike algebras and tree blocks studied by Lyle and Ruff.

Abstract

Let B be a weight-$3$ block of an Iwahori--Hecke algebra of type B over any field. We develop the combinatorics of B to prove that the decomposition numbers for B are all 0 or 1.

Decomposition numbers for weight 3 blocks of Iwahori--Hecke algebras of type B

TL;DR

The paper proves that in a weight block of an Iwahori--Hecke algebra of type , all decomposition numbers are in {0,1} over any field. The authors develop a detailed combinatorial framework based on bipartitions, the abacus, and restricted bipartitions, and they leverage Brundan--Kleshchev branching rules together with the cyclotomic Jantzen--Schaper formula to bound multiplicities. A careful block-by-block analysis, distinguishing core vs non-core blocks and four block types (II–IV), reduces the problem to a finite number of exceptional bipartitions whose contributions are controlled via JS-order arguments and duality. The results yield a complete, inductive proof that extends known type A weight- results to type and informs efficient computation of decomposition numbers via the Jantzen--Schaper formula, with implications for Ariki--Koike algebras and tree blocks studied by Lyle and Ruff.

Abstract

Let B be a weight- block of an Iwahori--Hecke algebra of type B over any field. We develop the combinatorics of B to prove that the decomposition numbers for B are all 0 or 1.
Paper Structure (33 sections, 12 theorems, 64 equations, 4 figures, 3 tables)

This paper contains 33 sections, 12 theorems, 64 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Basic notation
  4. Partitions and bipartitions
  5. Beta-sets and the abacus
  6. Restricted bipartitions
  7. Iwahori--Hecke algebras
  8. Blocks and weight
  9. Conjugation and component-switching
  10. Branching rules
  11. The cyclotomic Jantzen--Schaper formula
  12. Blocks of weight $3$
  13. Weight and the abacus
  14. The bipartitions in a non-core weight $3$ block
  15. Inductive approach to the main theorem
  16. ...and 18 more sections

Key Result

lemma 1

Suppose $\la$ is a restricted bipartition. Then both $\la^{(1)}$ and $\la^{(2)}$ are $e$-restricted partitions.

Figures (4)

  • Figure 1: Jantzen--Schaper interval for Cases \ref{['iii21']} and \ref{['iii22']}
  • Figure 2: Jantzen--Schaper interval for case \ref{['iv324']}
  • Figure 3: Jantzen--Schaper interval for case \ref{['iv12']}
  • Figure 4: Jantzen--Schaper interval for case \ref{['iv13']}

Theorems & Definitions (12)

  • lemma 1
  • lemma 2: mfweightLemma 3.6
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • lemma 7
  • lemma 8
  • lemma 9
  • lemma 10
  • ...and 2 more