Weyl submodules in restrictions of simple modules
arXiv:0904.0787 · doi:10.1016/j.jalgebra.2008.11.034
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(ω)\downarrow_{SL_{n-1}(F)}, where L(ω) is a simple SL_n(F)-module. This allows us to construct new primitive vectors in L(ω)\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.