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

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 $E_6$.