A local criterion for Weyl modules for groups of type A

TL;DR

This paper introduces a new local criterion and an algorithm to determine non-vanishing of certain vectors in Weyl modules for groups of type A, avoiding the use of bases and extending previous partial results.

Contribution

It proposes a novel algorithm for analyzing Weyl modules that is applicable to type A groups and proves the conjecture in both directions for these groups.

Findings

Algorithm determines if Fe^+_ e 0 in Weyl modules.

Conjecture proved for groups of type A.

Method avoids using bases of Weyl modules.

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.