Geodesic
Noncrossing sets and a Graßmannian associahedron
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014).

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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 $\binom{[n]}{k}$ 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, 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 $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc. 
@article{DMTCS_2014_special_265_a52,
     author = {Santos, Francisco and Stump, Christian and Welker, Volkmar},
     title = {Noncrossing sets and a {Gra{\ss}mannian} associahedron},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
     year = {2014},
     doi = {10.46298/dmtcs.2427},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2427/}
}
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%D 2014
%V DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
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Santos, Francisco; Stump, Christian; Welker, Volkmar. Noncrossing sets and a Graßmannian associahedron. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi : 10.46298/dmtcs.2427. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2427/

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