Polystability in positive characteristic and degree lower bounds for invariant rings

TL;DR

This paper introduces a new representation theoretic technique applicable in all characteristics to detect closed orbits, providing an algorithm for symmetric polynomials and establishing exponential degree lower bounds for invariant rings in key GCT actions.

Contribution

It develops a simplified, computationally efficient method for detecting closed orbits using Kempf's theory, and proves exponential lower bounds on invariant ring generators in important cases.

Findings

New algorithm to determine closed orbits of symmetric polynomials.

Exponential lower bounds on degrees of invariant ring generators.

Technique applicable in all characteristics.

Abstract

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf's theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to decide if a symmetric polynomial in $n$-variables has a closed ${\rm SL}_n$ orbit. As an important application, we prove exponential lower bounds on the maximal degree of a system of generators of invariant rings for two actions that are important from the perspective of Geometric Complexity Theory (GCT). The first is the action of ${\rm SL}(V)$ on ${\rm Sym}^3(V)^{\oplus 3}$, the space of 3-tuples of cubic forms, and the second is the action of ${\rm SL}(V) \times {\rm SL}(W) \times {\rm SL}(Z)$ on the tensor space $(V \otimes W \otimes Z)^{\oplus 5}$. In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.