The ring of Weyl invariant $E_8$ Jacobi forms

TL;DR

This paper establishes an isomorphism between the ring of Weyl invariant E8 Jacobi forms and covariants of binary forms, demonstrating its finite generation and providing a minimal basis of generators.

Contribution

It reveals a novel algebraic structure linking Weyl invariant Jacobi forms to classical covariants, and explicitly constructs a minimal generating set.

Findings

The ring of Weyl invariant E8 Jacobi forms is finitely generated.

An explicit minimal basis of generators is obtained.

The ring is isomorphic to the ring of joint covariants of binary sextic and quartic forms.

Abstract

We prove that the ring of Weyl invariant $E_8$ weak Jacobi forms is isomorphic to that of joint covariants of a binary sextic and a binary quartic form. The ring is therefore finitely generated. A minimal basis of generators is obtained from that already known for the ring of covariants.