NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.1017/fms.2017.1
· OA: W1585204912
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 et al. [ J. Algebra 324 (5) (2010), 951–969]. 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 Grassmann–Plücker 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 Grassmann associahedron . We extend results of Petersen et al. [ J. Algebra 324 (5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians 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 Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [ Amer. Math. Soc. Transl. 181 (2) (1998), 85–108]; see also Scott [ J. Algebra 290 (1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [ J. Algebra 324 (5) (2010), 951–969] but actually its cyclically invariant part.
