Some Weyl modules of the algebraic groups of type $E_6$

Let $G$ be a simple algebraic group of type $E_6$ over an algebraically closed field of characteristic $p>0$. We determine the submodule structure of the Weyl modul es with highest weight $r{\omega }_1$ for $0\le r\le p-1$, where ${\omega }_1$ is the fundamental weight of the standard $27$-dimensional module. In the process, the structures of other Weyl modules with highest weights linked to $r{\omega }_1$ are also found. %We also give some computations for the Weyl modules with highest weights %of the form $r({\omega }_1+{\omega }_6)$, which arise in the study of %the graph automorphism and associated twisted finite groups.