A local criterion for Weyl modules for groups of type A

Abstract

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 X_{\alpha,m} with \alpha<0. We conjecture an algorithm to determine if Fe^+_\omega\ne0, where F\in\U^-, \omega is a dominant weight and e^+_\omega is a highest weight vector of the Weyl module \Delta(\omega). This algorithm does not use bases of \Delta(\omega) 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.