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.
Mendeley helps you to discover research relevant for your work.
CITATION STYLE
Santos, F., Stump, C., & Welker, V. (2014). Noncrossing sets and a Graßmann associahedron. In Discrete Mathematics and Theoretical Computer Science (pp. 609–620). Discrete Mathematics and Theoretical Computer Science. https://doi.org/10.46298/dmtcs.2427