An implementation of the morphisms $SL_2(\mathbb{F}) \rightarrow SL_2(\mathsf{K}) \rightarrow \mathsf{X}$

TL;DR

The paper addresses the problem of explicitly realizing morphisms $sl_2(\mathbb{F}) \rightarrow sl_2(\mathsf{K}) \rightarrow \mathsf{X}$ within black box groups, by providing a GAP-based framework to construct ${\rm PGL}_2(\mathbb{F})$ and to map elements through a chain of groups ending at the black box group $\mathsf{X}$. The main method combines a diagonal automorphism, a semidirect product construction, and a change-of-basis between ${\rm SO}_3^{\flat}$ and ${\rm SO}_3^{\sharp}$, enabling explicit representations of $sl_2$ in a black box context, including decomposition into unipotents and Chevalley-commutator verification. Key contributions include the detailed GAP routines ${\sf SetUpForPGL2}$ and ${\sf ToolBoxSL2}$, the integration with a black box field $\mathsf{K}$ inside $\mathsf{X}$, and practical guidance with an example $X \vDash sl_2(997)$, together with notes on runtime considerations and an announced inverse morphism in BY2025. This work advances computational representation theory in black box settings, offering a concrete pipeline for researchers to realize and verify explicit morphisms in large finite groups.

Abstract

We briefly explain how to implement the morphisms in our paper ``Natural representations of black box groups encrypting $SL_2(\mathbb{F})$" and provide some examples.