On vertices of simple modules for symmetric groups of small degrees.
The purpose of the paper under review is to present some algorithmic approaches towards the determination of vertices of indecomposable modules over group algebras, most of which do indeed work for arbitrary finite groups. Furthermore, apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3, respectively, with one exception. Their computations have basically been carried out using the computer algebra system Magma [\textit{W. Bosma, J. Cannon, C. Playoust}, J. Symb. Comput. 24, No. 3-4, 235-265 (1997; Zbl 0898.68039)] and are based on algorithms which have been developed by the third author [in Vertizes einfacher Moduln der symmetrischen Gruppen, PhD thesis, Jena, \url{http://www.minet.unijena.de/algebra/preprints/preprints.html}, 2004] and have later been extended by the first author [in Theoretische und algorithmische Methoden zur Berechnung von Vertizes irreduzibler Moduln symmetrischer Gruppen, PhD thesis, Jena, \url{http://www.minet.uni-jena.de/algebra/preprints/preprints.html}, 2007]. Furthermore, one of the computationally investigated simple modules has been treated with the C-MeatAxe [\textit{M. Ringe}, The C MeatAxe release, user's guide and technical reference, \url{http://www.math.rwth-aachen.de~MTX/meataxe.ps} (1998)] and GAP [The GAP Group, GAP-4-Groups, Algorithms and Programming, Version 4.4.9 (2006)]. The present paper is structured as follows. In the first section they briefly summarize the basic facts about vertices of indecomposable \(FG\)-modules to provide the theoretical background needed for their computations. Moreover, they prove a result on vertices of simple \(FG\)-modules which involves the theory of twisted group algebras. Afterwards, in Section 3, they present their algorithmic methods. The tables in Appendix A of this article display their results on vertices and sources of simple modules for the symmetric groups.
scientific article; zbMATH DE number 5349027
scientific article