NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON
NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON
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DOI：
10.1017/fms.2017.1
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发表时间：
2014-03
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影响因子：
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通讯作者：
F. Santos;Christian Stump;V. Welker
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文献类型：
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作者：
F. Santos;Christian Stump;V. Welker
我们研究了$[n]$中元素对到$[n]$中$k$ -元组之间的非交叉关系的自然推广，这是Petersen等人首先考虑的[J. Algebra 324（5）（2010），951-969]。我们给出了一个替代的方法来证明他们的结果：由这个关系导出的$\binom{[n]}{k}$上的旗单纯复形是由两个链的乘积$[k]\times [n-k]$给出的偏序集的序多面体的正则、么模和旗三角剖分（也称为Gelfand-Tsetlin多面体），并且它是单形和球面的连接（即，它是Gorenstein三角剖分）。然后，我们观察到，这已经意味着存在一个旗单纯多面体推广的对偶结合面体，其斯坦利-赖斯纳理想是格拉斯曼-普吕克尔理想的初始理想，而以前的建设这样的多面体并不保证旗，也没有减少到对偶结合面体为$k=2$。在我们的方式中，我们提供一般结果的秩序多面体和他们的三角剖分。我们称单纯复形为非交叉复形，由它导出的多面体为对偶格拉斯曼结合面体。我们推广了Petersen等人[J. Algebra 324（5）（2010），951-969]的结果，证明了非交叉复形和格拉斯曼结合面自然地反映了具有不同参数的格拉斯曼之间的关系，特别是同构$G_{k，n}\cong G_{n-k，n}$ .此外，我们的方法使我们能够表明，邻接图的非交叉复杂承认一个自然的非循环方向，使我们能够定义一个格拉斯曼-塔马利秩序的最大非交叉家庭。最后，我们看看精确的关系的非交叉复杂性和弱可分性复杂的勒克莱尔和Zelevinsky [美国数学学会。181（2）（1998），85-108];另见Scott [J. Algebra 290（1）（2005），204-220]等。我们证明了弱可分性复形不仅是Petersen等人[J. Algebra 324（5）（2010），951-969]指出的非交叉复形的子复形，而且实际上是其循环不变部分。
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.