On Weyl and Tilting Modules for $G_{2}$ when $p=2$.

In this paper the authors investigate the structure of Weyl and tilting modules for the algebraic group $G_{2}$ over an algebraically closed field of characteristic $2$ that are related to projective indecomposable modules for the first Frobenius kernel. It is shown that there exists one projective indecomposable module for the first Frobenius kernel that is not an indecomposable tilting module which yields a counterexample to Donkinu0027s Tilting Module Conjecture. Counterexamples to other related conjectures are also presented.