Algebraic construction of Weyl invariant $E_8$ Jacobi forms

We study the ring of Weyl invariant $E_8$ weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant $E_8$ weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant $E_8$ weak Jacobi forms of given index $m$ for $m\le 20$. We also determine the lowest weight generators of the free module of index $m$ for $m\le 28$. Our results support the lower bound conjecture of Sun and Wang and prove explicitly that there exist generators of the ring of Weyl invariant $E_8$ weak Jacobi forms of weight $-4m$ and index $m$ for all $12\le m \le 28$.