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Intersections of blocks of cyclotomic Hecke algebras

Maria Chlouveraki, Gunter Malle

Abstract

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from $E_8$. We also propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.

Intersections of blocks of cyclotomic Hecke algebras

Abstract

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from . We also propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.

Paper Structure

This paper contains 20 sections, 19 theorems, 15 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Blocks in Harish-Chandra series
  3. Cyclotomic Hecke algebras
  4. The conjecture of Trinh and Xue
  5. Verifying the conjecture
  6. The cyclic case
  7. Ennola duality
  8. Algorithmic determination of blocks
  9. Blocks in the exceptional types
  10. Type $G_2$
  11. Type ${}^{3}\!D_4$
  12. Type $F_4$
  13. Type $E_6$
  14. Type $E_7$
  15. Type $E_8$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Conjecture conj:TX holds for all finite reductive groups of exceptional type, except possibly for type $E_8$ when $d\in\{3,4,6\}$. For $E_8$ and $d=3,4,6$, we have obtained approximate block decompositions that are in accordance with Conjecture conj:TX.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Conjecture 2.3: Trinh--Xue
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 28 more