  We study a natural generalization of the noncrossing relation between pairs
of elements in [n] to k-tuples in [n] that was first considered by Petersen,
Pylyavskyy, Speyer (2010). We give an alternative approach to their result that
the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a
regular, unimodular and flag triangulation of the order polytope of the poset
given by the product $[k]\times[n-k]$ of two chains (also called
Gelfand-Tsetlin polytope), and that it is the join of a simplex and a sphere
(that is, it is a Gorenstein triangulation). We then observe that this already
implies the existence of a flag simplicial polytope generalizing the dual
associahedron, whose Stanley-Reisner ideal is an initial ideal of the
Gra{\ss}mann-Pl\"ucker ideal, while previous constructions of such a polytope
did not guarantee flagness nor reduced to the dual associahedron for k=2. On
our way we provide general results about order polytopes and their
triangulations. We call the simplicial complex the noncrossing complex, and the
polytope derived from it the dual Gra{\ss}mann associahedron. We extend results
of Petersen, Pylyavskyy, Speyer (2010) showing that the non-crossing complex
and the Gra{\ss}mann associahedron naturally reflect the relations between
Gra{\ss}mannians with different parameters, in particular the isomorphism
$G_{k,n} \cong G_{n-k,n}$. Moreover, our approach allows us to show that the
adjacency graph of the noncrossing complex admits a natural acyclic orientation
that allows us to define a Gra{\ss}mann-Tamari order on maximal noncrossing
families. Finally, we look at the precise relation of the noncrossing complex
and the weak separability complex of Leclerc, Zelevinsky (1998) and show that
the weak separability complex is the cyclically invariant part of the
noncrossibg complex.
