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]. We show that the flag simplicial complex on ([nk]) induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product [k] × [n - k] of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism Gk,n~= Gn-k,n. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc.