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Generic representations of finite general linear groups in cross characteristic

Aurélien Djament, Thomas Gaujal

Abstract

We study generic representations of general linear groups over a finite ring R with coefficients in a field k in which the cardinality of R is invertible, that is functors from finitely-generated projective R-modules to k-vector spaces. We obtain especially a classification of such simple representations, what allows to prove a conjecture of Djament-Touz{é}-Vespa on dimensions taken by such a functor.

Generic representations of finite general linear groups in cross characteristic

Abstract

We study generic representations of general linear groups over a finite ring R with coefficients in a field k in which the cardinality of R is invertible, that is functors from finitely-generated projective R-modules to k-vector spaces. We obtain especially a classification of such simple representations, what allows to prove a conjecture of Djament-Touz{é}-Vespa on dimensions taken by such a functor.
Paper Structure (30 sections, 98 equations)

This paper contains 30 sections, 98 equations.

Table of Contents

  1. Représentations des catégories additives
  2. Foncteurs polynomiaux et antipolynomiaux
  3. Linéarisation des foncteurs additifs
  4. Morphismes
  5. Dualité ; applications
  6. Groupes d'extensions
  7. Les foncteurs ${\operatorname{Q}}_A$ et ${\operatorname{Q}}^A$
  8. Généralités
  9. Résolution fondamentale
  10. Groupes d'extensions
  11. Les foncteurs ${\operatorname{Q}}_{A,M}$ et ${\operatorname{Q}}^{A,M}$
  12. Les foncteurs ${\operatorname{Q}}(A,M)$
  13. Foncteurs de décalage et de différence paraboliques
  14. Les foncteurs $\bar{\tau}_x$
  15. Les foncteurs $\bar{\delta}_x$
  16. ...and 15 more sections

Theorems & Definitions (55)

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