On the computation of the nilpotent pieces in bad characteristic for algebraic groups of type G2, F4, and E6

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field 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 g called pieces. This approach appears in Lusztig's article from 2011. The pieces for the exceptional groups of type G2,F4,E6,E7, and E8 in bad characteristic have not yet been determined. This article presents a solution, relying on computational techniques, to this problem for groups of type G2, F4, and E6.