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

## TL;DR

This paper explicitly derives polynomial relations in the character ring of E8 involving exterior powers of the adjoint representation, with implications across various advanced mathematical and physical theories.

## Contribution

It provides explicit polynomial relations in the character ring of E8's exterior powers, solved via a finite linear problem and distributed computation, advancing understanding in representation theory.

## Key findings

- Explicit polynomial relations for E8 characters derived.
- Reduction to a finite linear problem enables efficient computation.
- Implications for integrable systems, Seiberg-Witten theory, and quantum topology.

## 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 $i^{\rm th}$ 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.