The structure of the algebra of weak Jacobi forms for the root system $F_4$

TL;DR

This paper proves that the algebra of Weyl-invariant weak Jacobi forms for the root system F4 is polynomial, extending previous work on similar structures for Dn root systems, and clarifies their algebraic structure.

Contribution

It establishes the polynomiality of the algebra of weak Jacobi forms for the F4 root system, expanding the understanding of their algebraic structure.

Findings

Proves polynomiality of the algebra of weak Jacobi forms for F4

Extends previous results from Dn root systems to F4

Provides a structural description of the algebra

Abstract

We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root systems of $D_n$ type for $2\leqslant n \leqslant 8$ was studied.