Untersuchungen zu Tensorprodukten von Moduln symmetrischer Gruppen
- On tensor products of modules of symmetric groups
Orlob, Johannes; Hiß, Gerhard (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2009)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2009
Tensor products of modules of a finite group occur in many areas of ordinary and modular representation theory. In general only little is known about the structure of these modules. This thesis is about various aspects of tensor products of modules of symmetric groups S_n over a field of prime characteristic p. The main focus lies on some special indecomposable summands of such tensor products and the complete decomposition into indecomposable summands of certain tensor products. Simple modules and trivial source modules are mainly focused. For instance we will look at tensor products of Young modules. We introduce the generalized sign sgn_n^q. We will show that this class function of S_n is a generalized character. We give an application of sgn_n^q to tensor products of Young modules. Additionally, we determine in some cases the set of irreducible constituents of sgn_n^q. We will although look at tensor products of the simple natural module and simple modules corresponding to hook partitions. We show that some of these products are semi-simple or indecomposable. Over a field of characteristic two we will determine the simple constituents of the tensor square of the basic spin module. And we will look at tensor products of simple modules lying in a block of cyclic defect or a block with defect group C_3 x C_3.