Noncrossing sets and a Graßmann associahedron

Abstract: 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.