正拟阵基本集的组合学
Combinatorics of Essential Sets for Positroids
AI总结：
本文将Fulton的基本集概念推广到有界仿射置换，利用其与正拟阵的双射关系研究二者联系，定义连通基本集和核，提出检索满足特定秩条件正拟阵的算法。
摘要：
正拟阵是Postnikov在研究非负格拉斯曼流形时引入的一类拟阵。Postnikov确定了若干与正拟阵双射对应的组合对象，其中包括有界仿射置换。另一方面，Fulton为置换引入的基本集概念，被Knutson用于研究区间秩正拟阵这一特殊家族。我们将Fulton的基本集推广到有界仿射置换。后者与正拟阵的双射关系使得我们能够研究它们之间的联系。从正拟阵的角度看，基本集是极大依赖循环区间。我们定义连通基本集，并证明它们给出正拟阵多面体的面描述，以及定义正拟阵簇的方程。我们定义基本集的一个子集称为核，它包含唯一恢复正拟阵所需的最小秩条件。我们提供一个算法，用于检索满足核中秩条件或循环区间上任何兼容秩条件的正拟阵。
英文摘要：
Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic interval. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
