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 on is a flag matroid that can be realized by a real matrix such that the minors of involving rows are nonnegative for all . In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety equals the nonnegative flag Dressian , and that the points of give rise to coherent subdivisions of the flag positroid polytope 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 -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 which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks is realizable.
中文翻译:
旗正类的多面体和热带几何
上的等级 的标志正仿体是一个标志拟阵,可以通过真实的 矩阵 实现,使得 <涉及行 的 的 b4> 次要元素对于所有 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 是连续数字序列时。在本例中,我们表明非负热带旗帜品种 等于非负旗帜 Dressian ,并且 的点 产生标志正类多胞体 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 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 ,它也是一个有向标志拟阵。然后我们证明每个正向的秩 的标志拟阵都是可实现的。