  We define the Weyl cycles on $X^n_s$, the blown up projective space
$\mathbb{P}^n$ in $s$ points in general position. In particular, we focus on
the Mori Dream spaces $X^3_7$ and $X^{4}_{8}$, where we classify all the Weyl
cycles of codimension two. We further introduce the Weyl expected dimension for
the space of the global sections of any effective divisor that generalizes the
linear expected dimension and the secant expected dimension.
