To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(đť”˝p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(đť”˝p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(đť”˝p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are ‘close’ to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
