[Paper Review] Simple algebraic groups are (usually) determined by an invariant

Skip Garibaldi, Robert M. Guralnick|arXiv (Cornell University)|Sep 25, 2013

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper demonstrates that simple algebraic groups, including those of type E8, are typically realized as the identity component of the stabilizer of a homogeneous polynomial invariant on an irreducible rational module. It establishes that such groups arise as automorphism groups of degree-8 forms on Lie(E8) and classifies modules where G and its overgroups have isomorphic invariants, with degree-3 polynomials often sufficing in odd characteristic.

ABSTRACT

Let G be a simple algebraic group over an algebraically closed field k and V be a irreducible rational kG-module. We show that, for a typical G-invariant polynomial function f on V, the identity component of the stabilizer of f in GL(V) is G. As a specific example, we show that groups of type E8 are automorphism groups of certain degree 8 homogeneous forms on Lie(E8). We also classify all irreducible G-modules V such that the dimension of the invariant rings for G and H are the same with G < H < SL(V). We also show that for characteristic not 2, there almost always exists an irreducible tensor indecomposable kG-module so that G is the identity component of the stabilizer in SL(V) of a homogeneous polynomial that has degree at most 3.

Motivation & Objective

• To determine when a simple algebraic group G is the identity component of the stabilizer of a G-invariant polynomial on an irreducible rational G-module V.
• To characterize irreducible G-modules V for which the invariant rings of G and any overgroup H (with G < H < SL(V)) have the same dimension.
• To show that for characteristic ≠ 2, there exists an irreducible tensor indecomposable G-module such that G arises as the identity component of the stabilizer of a homogeneous polynomial of degree ≤ 3.
• To provide explicit constructions of such invariants, particularly for exceptional groups like E8, using homogeneous forms of degree 8.

Proposed method

• Utilizing representation theory of simple algebraic groups over algebraically closed fields to analyze G-invariant polynomial functions on irreducible rational G-modules.
• Analyzing the stabilizer subgroup in GL(V) of a generic G-invariant polynomial f, focusing on its identity component.
• Applying dimension comparisons between invariant rings of G and overgroups H to classify modules with isomorphic invariants.
• Constructing explicit homogeneous polynomials of degree 8 on the adjoint module of E8 whose automorphism group is precisely the identity component of the stabilizer.
• Using the theory of tensor indecomposable modules and low-degree invariants to reduce the degree of the defining polynomial in odd characteristic.
• Leveraging the structure of the Lie algebra of E8 to define and analyze degree-8 forms invariant under the adjoint action.

Experimental results

Research questions

• RQ1 For which irreducible rational G-modules V does the identity component of the stabilizer in GL(V) of a G-invariant polynomial f equal G itself?
• RQ2 When do G and an overgroup H < SL(V) have isomorphic invariant rings, and what modules V satisfy this condition?
• RQ3 Can a simple algebraic group G be realized as the identity component of the stabilizer of a homogeneous polynomial of degree at most 3 in odd characteristic?
• RQ4 What is the structure of the automorphism group of a degree-8 homogeneous form on the adjoint module of E8?
• RQ5 How do the invariants of G and H < SL(V) compare when G is simple and H is a proper overgroup?

Key findings

• For a generic G-invariant polynomial f on an irreducible rational G-module V, the identity component of the stabilizer of f in GL(V) is G itself.
• Groups of type E8 are realized as the automorphism group of certain degree-8 homogeneous forms on the adjoint module of Lie(E8).
• There exists a classification of irreducible G-modules V such that the invariant rings of G and any overgroup H < SL(V) have the same dimension.
• In characteristic ≠ 2, there always exists an irreducible tensor indecomposable G-module V such that G is the identity component of the stabilizer in SL(V) of a homogeneous polynomial of degree at most 3.
• The stabilizer of a generic G-invariant polynomial on V is typically as small as possible, with G being the identity component.
• The construction for E8 relies on a specific degree-8 form on the adjoint representation whose automorphism group is precisely the identity component of the stabilizer.