We describe a new condensation method for computing the submodule lattice of a module for a finite dimensional algebra over a finite field, which exploits the idea of condensation and extends it to the case of primitive idempotents. The method has been implemented in the new C version of the Meat-Axe developed at Aachen, and we give several examples which have been analysed with our method.1991 Mathematics Subject Classification. 06C05, 15-04, 16G10, 20C40.
In many applications of representation theory of finite groups numerical computations for particular groups are called for. Although there are cases where one has to construct matrices for representations, in the majority of cases it is sufficient to work with characters, in fact this seems to be the only way to deal with many problems for larger groups. In order to deal effectively with characters of finite groups on computers, two computer systems CAS (Qharacter Algorithm System) and MOC (Modular Qharacters) have been developed in Aachen. CAS deals mainly with ordinary characters and had been designed by J. Neubiiser, H. Pahlings and W. Plesken and described in [NPP 84].Since then it has been substantially revised and extended, see section 1. MOC on the other hand deals with modular characters and was developed by G. HiB, K. Lux and R. Parker mainly to compute decomposition numbers for finite groups.The usefulness of the computer systems mentioned is greatly enhanced by the fact, that they contain large libraries of character tables, ordinary and modular ones. CAS now contains all the character tables published in the Atlas of Finite Groups [Con 85] and many others, in particular the character tables of many maximal subgroups of the sporadic simple groups (although completeness has not been achieved so far). The latter have proved particularly useful for many applications and for solving some open questions. We mention just one, raised independently by several people, cf. e.g. It should be mentioned that CAS and MOC are now closely linked together, so that in particular both libraries of character tables can be used by either system. The paper is organized as follows: in the first section the main new features and most important new algorithms of CAS concerning ordinary characters are described. Section 2 contains applications for analyzing the subgroup structure of a group, in particular an algorithm for computing the table of marks is presented. Also the Mobius functions of the lattice of subgroups and of the poset of conjugacy classes of subgroups are studied. In section 3 the main ideas and algorithms of MOC are outlined and an overview is given on the results achieved with it concerning the computation of decomposition numbers.Acknowledgement: We wish to thank the Deutsche Forschungsgemeinschaft for financial support.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.