Associated Forms in Classical Invariant Theory

TL;DR

This paper proves a conjecture that all classical invariants of forms of degree m in complex n can be derived from associated forms of degree n(m-2), confirming it for all cases and proposing a stronger version.

Contribution

The paper fully proves the conjecture for all degrees and dimensions, extending previous partial results and introducing a stronger conjectural framework.

Findings

Confirmed the conjecture in full generality.

Proposed a stronger version of the conjecture.

Provided evidence supporting the stronger conjecture.

Abstract

It was conjectured in a recent article by M. Eastwood and the second author that all absolute classical invariants of forms of degree $m\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of degree $n(m-2)$ by means of assigning every form with non-vanishing discriminant the so-called associated form. In that paper, this surprising conjecture was confirmed for binary forms of degree $m \le 6$ and ternary cubics. In the present article, we settle the conjecture in full generality. In addition, we propose a stronger version of this statement and obtain evidence supporting it.