Computations with nilpotent orbits in SLA

TL;DR

This paper reports computational results on nilpotent orbits in exceptional Lie algebras using the SLA package in GAP4, confirming classifications, answering open questions, and extending classical results.

Contribution

It provides computational verification of nilpotent orbit classifications, addresses open questions, and extends classical results to exceptional Lie algebras.

Findings

Confirmed classification of reachable nilpotent orbits in exceptional Lie algebras.

Answered a question by Panyushev regarding nilpotent orbits.

Extended Yakimova's results from classical to exceptional Lie algebras.

Abstract

We report on some computations with nilpotent orbits in simple Lie algebras of exceptional type within the SLA package of GAP4. Concerning reachable nilpotent orbits our computations firstly confirm the classification of such orbits in Lie algebras of exceptional type by Elashvili and Grelaud, secondly they answer a question by Panyushev, and thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types. The second topic of this note concerns abelianizations of centralizers of nilpotent elements. We give tables with their dimensions.