[Paper Review] Generic stabilizers in actions of simple algebraic groups II: higher Grassmannian varieties

Robert M. Guralnick, R. Lawther|arXiv (Cornell University)|Apr 30, 2019

Algebraic structures and combinatorial models4 citations

TL;DR

This paper investigates the generic stabilizers of simple algebraic groups acting on higher Grassmannian varieties—spaces of subspaces of fixed dimension greater than one in irreducible modules. Through explicit computation, it shows that except for one case, a dense open subset of each Grassmannian has stabilizer conjugate to a fixed subgroup, and provides tables of non-trivial generic stabilizers while determining the existence of dense orbits.

ABSTRACT

This is the second of two papers treating faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties; in the first paper we considered the module itself and its projective space, while here we handle the varieties comprising the subspaces of the module of fixed dimension greater than one. By explicit calculation, we show that in each case, with essentially one exception, 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.

Motivation & Objective

• To classify generic stabilizers for simple algebraic groups acting on Grassmannian varieties parameterizing subspaces of fixed dimension >1 in irreducible modules.
• To extend the results of the first paper—focused on modules and projective spaces—to higher-dimensional Grassmannians.
• To determine, for each case, whether a dense orbit exists under the group action.
• To provide explicit tables listing non-trivial generic stabilizers in the relevant cases.

Proposed method

• Explicit computation of stabilizers using the structure of irreducible representations and the action of simple algebraic groups on subspaces.
• Analysis of the group action on Grassmannian varieties via orbit-stabilizer theory and geometric invariant theory.
• Identification of a dense open subset in each Grassmannian where the stabilizer is conjugate to a fixed subgroup.
• Use of representation-theoretic techniques to classify the possible generic stabilizers across all cases.
• Comparison of the stabilizer structure with known results from the first paper on projective spaces.
• Systematic determination of whether the action admits a dense orbit by examining the orbit structure.

Experimental results

Research questions

• RQ1 For which irreducible modules and fixed dimensions d > 1 does the action of a simple algebraic group on the Grassmannian of d-dimensional subspaces have a dense open subset with conjugate stabilizers?
• RQ2 What are the explicit isomorphism types of the generic stabilizers in these actions, excluding the exceptional case?
• RQ3 Does the group action on each Grassmannian variety admit a dense orbit, and if so, under what conditions?
• RQ4 How do the generic stabilizers in higher Grassmannians compare to those in the projective space case studied in the first paper?
• RQ5 Which cases yield non-trivial generic stabilizers, and how can they be systematically tabulated?

Key findings

• In all cases except one, there exists a dense open subset of the Grassmannian where the stabilizer of any point is conjugate to a fixed subgroup, known as the generic stabilizer.
• The paper provides explicit tables listing the isomorphism types of non-trivial generic stabilizers for each case.
• For each action, the paper determines whether a dense orbit exists, with the answer being affirmative in most cases.
• The exceptional case—where the generic stabilizer is not conjugate to a fixed subgroup across a dense open set—is explicitly identified and characterized.
• The results extend the classification from projective spaces to higher Grassmannians, revealing a consistent pattern in stabilizer structure.
• The method of explicit computation successfully resolves the stabilizer structure even in complex representation-theoretic settings.