Computations of generating lengths with GAP

TL;DR

This paper presents a new computational method implemented in GAP for calculating the generating length in modular representation theory, enabling new insights and conjectures on ghost numbers of specific groups.

Contribution

It introduces a novel algorithm and implementation in GAP to compute generating lengths, advancing computational capabilities in modular representation theory.

Findings

Successfully computed examples previously infeasible

Formulated new conjectures on ghost numbers of Q_8 and A_4

Enhanced understanding of generating numbers in group algebras

Abstract

In this paper, we discuss how to apply GAP to do computations in modular representation theory. Of particular interest is the generating number of a group algebra, which measures the failure of the generating hypothesis in the stable module category. We introduce a computational method to do this calculation and present it in pseudo-code. We have also implemented the algorithm in GAP and managed to do computations of examples that we were not able to do before. The computations lead to conjectures on the ghost numbers of the groups $Q_8$ and $A_4$.