Weyl submodules in restrictions of simple modules

TL;DR

This paper investigates how Weyl modules over SL_{n-1}(F) can be embedded into the restrictions of simple modules over SL_n(F), enabling the construction of new primitive vectors in modular representation theory.

Contribution

It demonstrates that certain Weyl modules can be embedded into restricted simple modules, providing a method to generate primitive vectors in modular representations.

Findings

Weyl modules embed into restrictions of simple modules under certain conditions

New primitive vectors can be constructed from Weyl modules in restricted modules

Examples confirm the embedding and construction methods are effective

Abstract

Let F be an algebraically closed field of characteristic p>0. Suppose that SL_{n-1}(F) is naturally embedded into SL_n(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SL_{n-1}(F) can be embedded into the restriction L(\omega)\downarrow_{SL_{n-1}(F)}, where L(\omega) is a simple SL_n(F)-module. This allows us to construct new primitive vectors in L(\omega)\downarrow_{\SL_{n-1}(F)} from any primitive vectors in the corresponding Weyl modules. Some examples are given to show that this result actually works.