The modality of a Borel subgroup in a simple algebraic group of type $E_8$

TL;DR

This paper computes the modality of the action of a Borel subgroup on the Lie algebra of its unipotent radical for the exceptional algebraic group of type E8, completing the classification for all exceptional types.

Contribution

It provides the first known value of the modality for E8, using computer calculations to finalize the classification for all exceptional algebraic groups.

Findings

mod(B : u) for E8 is 20

Completes the classification for all exceptional types

Uses computer algebra for complex calculations

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}\, k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B : \mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $\mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $\mathrm{mod}(B:\mathfrak u) = 20$, when $G$ is of type $\mathrm E_8$. This completes the determination of the values for $\mathrm{mod}(B:\mathfrak u)$ for $G$ of exceptional type.