On the computation of the nilpotent pieces in bad characteristic for algebraic groups of type $G_2$, $F_4$, and $E_6$

TL;DR

This paper computes the nilpotent pieces in the unipotent variety of algebraic groups of types G2, F4, and E6 in bad characteristic, using computational methods to fill a gap in the existing classification.

Contribution

It provides the first explicit determination of nilpotent pieces for G2, F4, and E6 in bad characteristic, extending Lusztig's framework with computational techniques.

Findings

Explicit descriptions of nilpotent pieces for G2, F4, E6

Computational methods applied to complex algebraic group problems

Fills gaps in the classification of unipotent varieties in bad characteristic

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\mathbf{k}$, and let Lie$(G)$ be its associated Lie algebra. In his series of papers on unipotent elements in small characteristic, Lusztig defined a partition of the unipotent variety of $G$. This partition is very useful when working with representations of $G$. Equivalently, one can consider certain subsets of the nilpotent variety of Lie$(G)$ called pieces. This approach appears in Lusztig's article from 2011. The pieces for the exceptional groups of type $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$ in bad characteristic have not yet been determined. This article presents a solution, relying on computational techniques, to this problem for groups of type $G_2$, $F_4$, and $E_6$.