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Generic stabilizers in actions of simple algebraic groups

R. M. Guralnick, R. Lawther

TL;DR

This work provides a comprehensive treatment of generic stabilizers for faithful actions of simple algebraic groups on irreducible modules and on Grassmannians. By combining weight-string analysis, unipotent-class closures, and codimension bounds (via Premet-type arguments and refined $p$-dependent variants), the authors establish that a generic stabilizer exists in essentially all cases, with a single characteristic-2 exception requiring a semi-generic stabilizer. The results are organized into six tables, cataloguing nontrivial generic stabilizers and clarifying when dense or regular orbits occur; key implications relate the finiteness/emptiness of stabilizers to the existence of dense orbits and to invariants in representation theory and algebraic group actions. The methodology unites classical root-system techniques with modern computational data (Lübeck’s Lubdata) and a structured framework of subsystems and dominant weights, enabling a systematic, case-by-case classification across both classical and exceptional groups. The findings advance invariant-theoretic and geometric understanding of group actions in arbitrary characteristic and have potential applications to invariants, cohomology, and essential dimension problems. Math notation is consistently employed, and all formulas are presented within $...$ delimiters.

Abstract

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception (only in characteristic 2), there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

Generic stabilizers in actions of simple algebraic groups

TL;DR

This work provides a comprehensive treatment of generic stabilizers for faithful actions of simple algebraic groups on irreducible modules and on Grassmannians. By combining weight-string analysis, unipotent-class closures, and codimension bounds (via Premet-type arguments and refined -dependent variants), the authors establish that a generic stabilizer exists in essentially all cases, with a single characteristic-2 exception requiring a semi-generic stabilizer. The results are organized into six tables, cataloguing nontrivial generic stabilizers and clarifying when dense or regular orbits occur; key implications relate the finiteness/emptiness of stabilizers to the existence of dense orbits and to invariants in representation theory and algebraic group actions. The methodology unites classical root-system techniques with modern computational data (Lübeck’s Lubdata) and a structured framework of subsystems and dominant weights, enabling a systematic, case-by-case classification across both classical and exceptional groups. The findings advance invariant-theoretic and geometric understanding of group actions in arbitrary characteristic and have potential applications to invariants, cohomology, and essential dimension problems. Math notation is consistently employed, and all formulas are presented within delimiters.

Abstract

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception (only in characteristic 2), there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

Paper Structure

This paper contains 33 sections, 253 theorems, 950 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Notation
  4. Weights and module structure
  5. Unipotent classes
  6. Preliminary results
  7. Triples having TGS
  8. Conditions implying TGS
  9. Criteria involving bounds for codimensions
  10. Relevant subsystems and dominant weights
  11. Exclusion of triples
  12. Weight string analysis
  13. Further analysis
  14. The triples $(C_4, \omega_3, 3)$ and $(B_2, \omega_1 + \omega_2, 5)$
  15. Tensor products
  16. ...and 18 more sections

Key Result

Theorem 1

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and $V$ a non-trivial irreducible $G$-module of dimension $d$.

Theorems & Definitions (485)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • Corollary 9
  • ...and 475 more