It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the fundamental weights are irreducible if the group is either $SL\_n$ or $SO\_n$ (Wong 71). For the symplectic group, this is not true for an arbitrary fundamental weight. Foulle provides certain conditions on the weights satisfying which the corresponding Weyl modules of the symplectic group are irreducible. Does one know anything about the irreducibility of the fundamental Weyl modules for the exceptional groups? I believe its true for $G\_2$ by results of Premet, Humphreys and others. Is it true also for other exceptional groups?



Irreducibility of fundamental Weyl modules
It is true in types $G\_2$, $F\_4$ and $E\_6$, but in types $E\_7$ and $E\_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E\_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in Progress in Math., vol. 95, 1991 (the best way to find this paper is to google the title). For related results, see also the paper by Gilkey and Seitz titled "Some representations of exceptional Lie algebras" and published in Geom. Dedicata, 25, (1988), 407-416.


That the Weyl module $V(\varpi\_6)$ for $G=E\_7(K)$ is reducible in characteristic $7$ is quite easy to see directly: the minuscule module $V(\varpi\_7)$ has a nondegenerate 
$G$-invariant symplectic form and by using Weyl's formula one observes that $V(\varpi\_6)$ can be obtained from the second fundamental Weyl module for $Sp\_{56}(K)$ by restriction to $G$. The latter Weyl module is reducible in characteristic $7$ as $7$ divides $56$. 


