What is the smallest simplest(non-trivial) $E\_8$ -module ?



smallest simplest $E_8$ -module 
Cartan showed that the lowest dimensional (nontrivial) $E\_8$-module is ${\frak{e}}\_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible module is considerably larger dimension, $3875$, and I think that the next one after that has dimension $30380$. 


At 
<http://www-math.univ-poitiers.fr/~maavl/LiE/form.html>, you can check these online using LiE.


