Tackling the generation problem in condensation

A major drawback of applying condensation in computational representation theory is the so-called generation problem, i.e. given a set of elements for the group algebra we generally do not know whether the corresponding condensed elements generate the condensed algebra. In this note we present two new methods to bridge this gap. Firstly we introduce generating sets which are in practice often small enough to be of computational use. Secondly we give a criterion which allows us to verify that a subset of the condensed algebra is in fact a generating set.