A computational study of certain Weyl modules for type $G_2$ in characteristic 2

Using the $\mathtt{\text{WeylModules}}$ $\mathsf{\text{GAP}}$ Package, we compute structural information about certain Weyl modules for type $G_2$ in characteristic $2$. This gives counterexamples to two conjectures stated by S.~Donkin in 1990. It also illustrates capabilities of the package, which can in principle be applied to Weyl modules for any simple, simply-connected algebraic group in any characteristic, subject of course to time and space limitations of computational resources.