On stable modular plethysms of the natural module of $\textrm{SL}_2(\mathbb{F}_p)$ in characteristic $p$

Abstract: 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 $k\textrm{SL}_2(\mathbb{F}_p)$ modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of $k\textrm{SL}_2(\mathbb{F}_p)$ 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 $k\textrm{SL}_2(\mathbb{F}_p)$-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.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.