I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf cohomology $H^i(\Omega\_{Gr(m,n)}^j$)?



Elementary way to compute Hodge numbers of Grassmanian
One of the ways is to use the projective bundle theorem that says that if $X = \mathbb{P}\_Y(E)$ is a projectivization of a rank $r$ vector bundle $E$ over $Y$ then
$$
H^\bullet(X) = H^\bullet(Y) \oplus H^\bullet(Y)[-2] \oplus \dots \oplus H^\bullet(Y)[2-2r],
$$
where $[s]$ stands for the shift of grading.


Now consider the partial flag variety with the natural projection
$$
p \colon Fl(1,2,\dots,k;n) \to Gr(k,n).
$$
On the one hand, $Fl(1,2,\dots,k;n)$ has a structure of an iterated projective bundle, hence all of its cohomology is of $(p,p)$-type, and its generating function is
$$
h\_{Fl}(t) = (1+t+\dots+t^{n-1})(1+t+\dots+t^{n-2})\cdots(1+t+\dots+t^{n-k}).
$$
On the other hand, the map $p$ is also an iterated projective bundle, hence $H^\bullet(Gr(k,n))$ is a direct summand of $H^\bullet(Fl(1,2,\dots,k;n))$, hence it is also of $(p,p)$-type, and moreover its generating function is
$$
h\_{Gr}(t) = \frac{h\_{Fl}(t)}{(1+t)(1+t+t^2)\cdots(1+t+ \dots+t^{k-1})}.
$$


