Computing component groups of stabilizers of nilpotent orbit representatives

TL;DR

This paper develops computational algorithms to explicitly determine the component groups of stabilizers of nilpotent elements in simple Lie algebras, overcoming limitations of previous hand calculations and enabling broader applicability.

Contribution

It introduces new algorithms for computing component groups of nilpotent orbit stabilizers, applicable to both classical and exceptional Lie algebra types, independent of prior isomorphism knowledge.

Findings

Algorithms successfully compute component groups for various nilpotent orbits.

Methods are implemented in the GAP computational algebra system.

Approach handles both classical and exceptional Lie algebra cases.

Abstract

The theory of nilpotent orbits of simple Lie algebras has seen tremendous developments over the past decades. In this context an important role is played by the component group of the stabilizer of a nilpotent element. In this work, the aim is to show computational methods to obtain explicit generators of the component group of the centralizer of a nilpotent element in a simple Lie algebra over $\mathbb{C}$. In some cases such generators had already been determined by impressive hand calculations but these often use the specific form of the chosen representative and are thus not immediately applicable to different representatives of the same orbit. It is then interesting to show how to overcome this issue constructing specific algorithms: for the classical types there is a straightforward method that directly translates well-known theoretical constructions; for the exceptional types we devise a method using the double centralizer of an $\mathfrak{sl}_2$-triple. In particular, it gives an independent construction of the component group, that does not depend on the prior knowledge of its isomorphism type. For our purposes, we needed to construct many algorithms which have been mainly implemented in the computational algebra system GAP.