Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unitriangularity of decomposition matrices of the unipotent ${\ell}$-blocks for simple adjoint exceptional groups

Marie Roth

Abstract

In 2020, Brunat-Dudas-Taylor showed that the decomposition matrix of unipotent ${\ell}$-blocks of a nite reductive group in good characteristic has unitriangular shape, under some conditions on the prime ${\ell}$, in particular ${\ell}$ being good. We extend this result to ${\ell}$ bad by adapting their proof to include the ${\ell}$-special classes dened by Chaneb.

Unitriangularity of decomposition matrices of the unipotent ${\ell}$-blocks for simple adjoint exceptional groups

Abstract

In 2020, Brunat-Dudas-Taylor showed that the decomposition matrix of unipotent -blocks of a nite reductive group in good characteristic has unitriangular shape, under some conditions on the prime , in particular being good. We extend this result to bad by adapting their proof to include the -special classes dened by Chaneb.
Paper Structure (42 sections, 37 theorems, 115 equations, 6 tables)

This paper contains 42 sections, 37 theorems, 115 equations, 6 tables.

Table of Contents

  1. Counting the number of modular representations of $G$
  2. Parametrisation of the ordinary characters of $G$
  3. Basic sets and their parametrisation
  4. The general approach for finding a unitriangular basic set
  5. Generalised Gelfand--Graev characters
  6. Unipotent conjugacy classes and nilpotent orbits
  7. Step 1: Parameterizing unipotent conjugacy classes reduces to parameterizing nilpotent orbits
  8. Step 2: Nilpotent orbits are parametrized by weighted Dynkin diagrams
  9. Definition of the generalised Gelfand--Graev characters
  10. Wave front set
  11. Admissible coverings and Kawanaka characters
  12. Definition and existence of an admissible covering
  13. Definition of a Kawanaka character
  14. Decomposition of the Kawanaka characters
  15. Character sheaves and characteristic functions
  16. ...and 27 more sections

Key Result

Theorem 1

Let $\mathbf{G}$ be a connected reductive group over ${k}$, an algebraically closed field of characteristic $p > 0$. Let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism endowing $\mathbf{G}$ with an $\mathbb{F}_q$-structure, for $q$ a power of the prime $p$. Assume the following: Then the $\ell$-decomposition matrix of the unipotent $\ell$-blocks of $\mathbf{G}^F$ is lower-unitriangul

Theorems & Definitions (77)

  • Conjecture : Geck
  • Theorem : Brunat--Dudas--Taylor
  • Theorem
  • Corollary
  • Theorem 1.1: geckBasicSetsBrauer1991
  • Definition 1.2: Chaneb
  • Theorem 1.3: chanebBasicSetsUnipotent2021
  • Proposition 2.1: geckBasicSetsBrauer1994
  • Definition 2.2
  • Proposition 2.3: Springer, Serre
  • ...and 67 more