  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.
