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Polyhedral and tropical geometry of flag positroids
Algebra & Number Theory ( IF 0.9 ) Pub Date : 2024-06-13 , DOI: 10.2140/ant.2024.18.1333
Jonathan Boretsky , Christopher Eur , Lauren Williams

A flag positroid of ranks r := (r1 < < rk) on [n] is a flag matroid that can be realized by a real rk × n matrix A such that the ri × ri minors of A involving rows 1,2,,ri are nonnegative for all 1 i k. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when r := (a,a + 1,,b) is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl r,n0 equals the nonnegative flag Dressian FlDr r,n0, and that the points μ = (μa,,μb) of TrFl r,n0 = FlDr r,n0 give rise to coherent subdivisions of the flag positroid polytope P(μ¯) into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its ( 2)-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids (χ1,,χk) which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks r = (a,a + 1,,b) is realizable.



中文翻译:


旗正类的多面体和热带几何



[n] 上的等级 r := (r1 < < rk) 的标志正仿体是一个标志拟阵,可以通过真实的 rk × n 矩阵 A 实现,使得 <涉及行 1,2,,riA 的 b4> 次要元素对于所有 1 i k 都是非负的。在本文中,我们探讨了旗正类的多面体和热带几何形状,特别是当 r := (a,a + 1,,b) 是连续数字序列时。在本例中,我们表明非负热带旗帜品种 TrFl r,n0 等于非负旗帜 Dressian FlDr r,n0 ,并且 TrFl r,n0 = FlDr r,n0 的点 μ = (μa,,μb) 产生标志正类多胞体 P(μ¯) 到标志正类多胞体的连贯细分。我们的结果适用于 Bruhat 区间多胞形:例如,我们表明,完整的旗拟阵多胞形是 Bruhat 区间多胞形当且仅当其 ( 2) 维面是 Bruhat 区间多胞形。我们的结果也适用于可实现性问题。我们将正向标志拟阵定义为一系列正向拟阵 (χ1,,χk) ,它也是一个有向标志拟阵。然后我们证明每个正向的秩 r = (a,a + 1,,b) 的标志拟阵都是可实现的。

更新日期:2024-06-13
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