Exterior powers of the adjoint representation and the Weyl ring of $E_8$

I derive explicitly all polynomial relations in the character ring of $E_8$ of the form $\chi_{\wedge^k \mathfrak{e}_8} - \mathfrak{p}_{k} (\chi_{1}, \dots, \chi_{8})=0$, where $\wedge^k \mathfrak{e}_8$ is an arbitrary exterior power of the adjoint representation and $\chi_{i}$ is the $i^{\rm th}$ fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg-Witten theory, quantum topology, orbifold Gromov-Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.