论文标题
加权刀片布置和积极的热带硕士
Weighted blade arrangements and the positive tropical Grassmannian
论文作者
论文摘要
在本文中，我们继续研究刀片布置和由$δ_{k，n} $引起的阳性细分。刀片是一种热带超浮面，它是由$ n $仿射简单的$ sl_n $根源的系统生成的，该系统具有循环对称性。当放置在单纯形的中心时，刀片会诱导分解为$ n $最大细胞，被称为Pitman-Stanley Polytopes。 我们介绍了一个复杂的$（b_ {k，n}，\ partial）$加权刀片的布置，我们证明了积极的热带格拉斯曼尼亚人向综合体的顶部组成部分，因此，face $δ__{2，n-（k-2）} $ untive y lime undect y yes n and（n n eys n and（1）（1（1））（1（k-1）分开。 最终，我们为所有超模拟的基本加权刀片安排介绍了层次结构，这些刀片在边界地图$ \ partial $下最少关闭，并应用我们的结果将同构型分类为同构类型的所有热带热带格拉斯曼尼亚$ \ text {trop} {trop} _+ g（3，n）$ n $ n $ n \ n $ n $ n \ le 9 $。
In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on $Δ_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type $SL_n$ that enjoys a cyclic symmetry. When placed at the center of a simplex, a blade induces a decomposition into $n$ maximal cells which are known as Pitman-Stanley polytopes. We introduce a complex $(B_{k,n},\partial)$ of weighted blade arrangements and we prove that the positive tropical Grassmannian surjects onto the top component of the complex, such that the induced weights on blades in the faces $Δ_{2,n-(k-2)}$ of $Δ_{k,n}$ are (1) nonnegative and (2) their support is weakly separated. We finally introduce a hierarchy of elementary weighted blade arrangements for all hypersimplices which is minimally closed under the boundary maps $\partial$, and apply our result to classify up to isomorphism type all rays of the positive tropical Grassmannian $\text{Trop}_+ G(3,n)$ for $n\le 9$.
