Tackling the generation problem in condensation.
`Condensation' is an important computational tool in the modular representation theory of finite groups, which permits computation of properties of very large representations of groups by reducing the problem to computing with much smaller representations of suitable `condensation algebras'. The major problem with such methods, however, is the difficulty of finding a tractable set of elements in the algebra which can be proved to generate the algebra. In this paper, the author makes two important advances in this area. The first is a method using double cosets of the normaliser of the condensation subgroup to produce a relatively small set of elements which provably generate the algebra. The second is a method for testing whether a given set of elements does in fact generate the algebra. Both techniques can be used to prove that the small (algebra) representations have exactly the same structures as the large (group) representations, and thus have practical applications to determining (for example) degrees and Brauer characters of irreducible representations. The most spectacular applications are to sporadic simple groups.
scientific article; zbMATH DE number 5141739