Fundamental invariants of orbit closures

TL;DR

This paper introduces a fundamental SL-invariant function for key objects in geometric complexity theory, analyzing their orbit closures, stabilizers, and polystability, with explicit formulas and computational evidence.

Contribution

It defines and studies a new invariant function for several important tensors and forms, generalizing classical invariants and linking to combinatorial conjectures.

Findings

Explicit invariant formulas for power sums in most cases

Identification of stabilizers and proof of polystability for all objects

Computer calculations supporting theoretical results

Abstract

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon-Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones).