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

Abstract

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 $i^{\mathrm{th}}$ 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.

v2: typos fixed. v3: several typos fixed, a gap in the proof of Lemma 2.1 has been filled, with the key argument of the proof now streamlined and strengthened; Appendix C eliminated in favour of a link to online data; 38 pages, 2 figures, 2 appendices, version accepted on J. Algebra. Ancillary binary files described in Appendix A available at http://tiny.cc/E8Char