A Local Criterion for Weyl Modules for Groups of Type A

Let G be a universal Chevalley group over an algebraically closed field and U− be the subalgebra of Dist(G) generated by all divided powers Xα,m with α < 0. We conjecture an algorithm to determine if Fe + ω 6= 0, where F ∈ U −, ω is a dominant weight and eω is a highest weight vector of the Weyl module ∆(ω). This algorithm does not use bases of ∆(ω) and is similar to the algorithm for irreducible modules that involves stepwise raising the vector under investigation. For an arbitrary G, this conjecture is proved in one direction and for G of type A in both.