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Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable.
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Excluded minors of interval positroids that are paving matroids
本文研究区间正拟阵的排除子式，证明铺砌情形可通过松弛相关超平面归约为三个基本族，并据此完成非正拟阵情形的分类，同时给出铺砌正拟阵情形的判定准则。
Positroid envelopes and graphic positroids
本文通过显式图构造证明每个positroid包络类含图拟阵，并证明图论positroid在包络类中唯一，且可由非负子式符号关联矩阵表示。
A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the $123$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as a disjoint union of matroids, …
Positroidal aspects of non-nesting rook placements
本文从positroid视角研究车拟阵的结构，用Grassmann项链刻画车拟阵，回答Lam的问题，并给出新证明及循环平坦类子类。
Flag matroids are a rich family of Coxeter matroids that can be characterized using pairs of matroids that form a quotient. We consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorial representations to explore characterizations of flag positroids. Given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that form quotients with it. We state this in terms of their associated decorated permutations. In proving our characterization we also fully describe the circuits of this family.
Characterizing positroid quotients of uniform matroids
本文研究两步flag positroid，完整刻画了包含一致拟阵的情形，证明Grassmann项链包含关系对一般flag positroid不充分，并解决了装饰排列通过循环移位相关的猜想。
A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.
Ternary and quaternary positroids
该研究给出三元与四元正拟阵的禁用子式刻画，明确三元正拟阵的构造方式与四元正拟阵的禁用子式，并完整刻画三元正拟阵的格拉斯曼项链纤维结构。
The Rayleigh property is a negative correlation property of electrical networks, which was generalized to matroids by Choe and Wagner. We prove that positroids, a class of matroid introduced by Postnikov which have seen many recent applications in both math and physics, have the Rayleigh property.
Excluding a line from positroids
本文确定了不含U_{2,ℓ+2}作为子式的简单秩r正阵胚的最大元素数，并刻画了达到最大值的结构，推广到双圆、格路、多路和colaminar等类。
Ehrhart positivity for paving positroids
本文证明任意铺砌正拟阵的基多面体具有正埃哈特系数，为费罗尼等人提出的正拟阵均为埃哈特正的核心猜想提供了新的正例与重要证据。
A well-known conjecture of Stanley is that the h-vector of any matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. Positroids are special class of linear matroids that play a crucial role in the field of total positivity. In this short note, we prove that Stanley's conjecture holds for positroids.
A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product…
The Ehrhart $h^*$-polynomials of positroid polytopes
该研究针对正广面多胞形的Ehrhart $h^*$-多项式展开，给出了基于排列下降的任意正广面多胞形$h^*$-多项式显式公式，推广了此前针对超单纯形的相关结果。
A positroid is a matroid defined by Postnikov to study the cells in the non-negative part of the Grassmannian. They are in bijection with decorated permutations. We show a way to explain contraction and restriction of positroids in terms of decorated permutations.
In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of $(a,b)$-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of s…
Schubert matroids, Delannoy paths, and Speyer's invariant
本文通过枚举Delannoy路径并引入新的基统计量，给出了Schubert拟阵的Speyer $g$-多项式的组合计算公式，揭示了其正性性质，并推广到任意拟阵的计算算法。
Sparse Paving Positroids
本研究利用Le-图、装饰排列与格拉斯曼项链分类稀疏铺砌正拟阵，完成其计数并建立与黄金比例相关序列、卢卡斯数的联系。
A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.
Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) correspond to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.
The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from previous ones. Matroids are combinatorial objects that generalize the notion of linear independence and have connections with other fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, …
Vakil studied the intersection theory of Schubert varieties in the Grassmannian in a very direct way: he degenerated the intersection of a Schubert variety X_mu and opposite Schubert variety X^nu to a union {X^lambda}, with repetition. This degeneration proceeds in stages, and along the way he met a collection of more complicated subvarieties, which he identified as the closures of certain locally closed sets. We show that Vakil's varieties are _positroid varieties_, which in particular shows they are normal, Cohen-Macaulay, have rational singularities, and are defined by the vanishing of Plüc…
A basis shape locus takes as input data a zero/nonzero pattern in an $n \times k$ matrix, which is equivalent to a presentation of a transversal matroid. The locus is defined as the set of points in the Grassmannian of $k$ planes in $\mathbb{R}^n$ which are the row space of a matrix with the prescribed zero/nonzero pattern. We show that this locus depends only on the transversal matroid, not on the specific presentation. When a transversal matroid is a positroid, the closure of its basis shape locus is the associated positroid variety. We give a sufficient, and conjecturally necessary, conditi…
Schubert coefficients of sparse paving matroids
本文将Grassmannian中Chow类在Schubert基下展开的系数定义为拟阵的Schubert系数，计算了非连通拟阵及所有稀疏铺砌拟阵的该系数，并证实了其非负性。
Tropical matroid Schubert varieties and the graded Möbius algebra
本文引入热带Matroid Schubert变种，证明其热带上同调环与Matroid的分级Möbius代数同构，扩展了Arrangement Schubert变种到任意Matroid。
Positroids, Dressian and stable polynomials
本文研究 positroids 与 Dressian、Lorentzian 多项式之间的关系，证明可关联多仿射齐次稳定多项式，并在更广义的 Rayleigh 拟阵上验证 Brändén–Huh 猜想。
A characterization of positroids, with applications to amalgams and excluded minors
本文给出拟阵胚的一个新刻画，并利用它证明两个拟阵胚在交集为独立克隆集时其自由融合仍为拟阵胚，同时构造出拟阵胚类的多参数无限族排除子式。
Rado matroids and a graphical calculus for boundaries of Wilson loop diagrams
本文提出广义Wilson圈图及其图形演算，直接刻画N=4超杨-米尔斯理论中正定拟阵的边界，并给出计算Grassmann项链的算法与Python实现。
Microlocal multiplicity of matroid Schubert varieties
本研究聚焦拟阵舒伯特簇相交复形的特征闭链重数，证明其为组合不变量且可由显式公式计算，并提出任意拟阵下该推广结果非负的猜想。
The Chow and augmented Chow polynomials of uniform matroids
本文通过给出均匀拟阵的Chow多项式与增广Chow多项式的显式组合公式证明Ferroni猜想，改进现有公式并基于舒伯特拟阵解释系数，还副产品计数了按秩、环数和余girth分类的舒伯特拟阵。
Polyhedral and Tropical Geometry of Flag Positroids
本文研究连续秩旗拟正构体的热带与多面体结构，证明非负热带旗簇等于非负旗 Dressian，并应用于 Bruhat 区间多面体判定及正向定向旗拟阵可实现性。
Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
本文比较了偏旗簇中开放投影Richardson簇与仿射Schubert胞腔的Segre--MacPherson类，并在Grassmannian情形下，利用仿射排列的pipe dreams显式构造了表示开放positroid簇该类的对称函数。
Tutte polynomials of matroids as universal valuative invariants
本文完整分类了使Tutte多项式成为泛赋值不变量的对偶与子式封闭拟阵族，并由此导出拟阵Tutte多项式的新关系，证明其可唯一表示为图Schubert拟阵Tutte多项式的整系数组合。
Total positivity for matroid Schubert varieties
本文定义线性子空间的全非负拟阵舒伯特簇，证明其为正则 CW 复形且同胚于闭球，并由此给出实拟阵舒伯特簇的正则 CW 结构。
Lattice path matroids and quotients
本文通过图刻画格路径拟阵的商拟阵，证明其构成秩多项式系数为Narayana数的分次偏序集，并证明全格路径旗拟阵对应于非负旗簇中的点，进而证明了Mcalmon、Oh和Xiang关于正环面拟阵商拟阵的猜想在格路径拟阵情形下成立。
The polytope of all matroids
本文将拟阵多面体表示为 Schubert 拟阵多面体指示函数的整数组合，研究了所有格点凸包的顶点即极值拟阵，证明著名拟阵类构成其面并确定维数，还发现了在可表示拟阵上非负但一般不成立的赋值不变量。
Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections
本文针对线丛格拉斯曼流形Gr≥0(2,n)提出正子的图论刻画，用于计算正子胞腔的维数、边界及交集，并提供算法求取给定集合中的最大正子。
Richardson varieties, projected Richardson varieties and positroid varieties
本文综述了Richardson簇与positroid簇的组合学、坐标环、参数化及分解等已知成果，并提出了开Richardson簇坐标环是UFD等原创结果。
When is the matroid Schubert variety $\mathbb{Q}$-Gorenstein?
该研究针对拟阵舒伯特簇$Y_V$，运用运算乔伊上同调证明其上线丛的限制性质，给出其为戈伦斯坦与$\funnybb{Q}$-戈伦斯坦的组合学刻画，并构造了三类不同性质的实例。
Derived Categories of Permutahedral and Stellahedral Varieties
本研究利用凝聚-可构造对应构造环面簇导出范畴中由nef线丛构成的满强例外系，将其应用于排列多面体等三类多面体对应的环面簇，并证明该例外系具有拟阵论描述、对称性及细化尖点半正交分解等良好性质。
Ehrhart positivity for lattice path matroids
本文证明了所有格路 matroid 都具有 Ehrhart 正性，统一并推广了过去二十年中关于 matroid 的 Ehrhart 正性的多个结果。作者基于之前关于 fence 的 order polynomial 正性的工作，主要结果支持 Ferroni, Jochemko 和 Schröter (2022) 关于 positroid 的 Ehrhart 正性的猜想，并进一步证明所有 Schubert matroid 都具有 Ehrhart 正性，从而解决了 Fan 和 Li (2024) 的猜想，并支持 Monical, Tokcan 和 Yong (2019) 关于 Schubitopes 的 Ehrhart 正性猜想。
Parametrization of positroid cells by Dubrovin-Natanzon Cartier divisors on maximal Mumford curves
该研究证明全正Schubert胞是对应Le图的极大Mumford曲线雅可比簇的实分支，并构造了正合胞点与该曲线上Dubrovin-Natanzon除子间的双射。
Chow classes of matroids and standard Young tableaux
本文通过研究蛇形矩阵的Chow类，将之与特定的ribbon Schur函数关联，并提供了Schubert基中系数的显式公式，同时简化了经典结果并证明了某些Schubert系数的正性。
Motivic Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
本文研究了开放投影里奇变种和仿射Schubert单元的motivic Chern类，通过递推关系和格拉斯曼流形中的正体种公式，建立了与扭曲Kazhdan–Lusztig R多项式的关系。
Hyperplane Arrangements and Compactifications of Vector Groups
本文研究了超平面配置的 Schubert 簇作为仿射空间等变紧化的充分必要条件，将理论推广至部分紧化及态射，揭示了其与环面簇和多面体扇对应关系的相似性。
The combinatorial geometry of particle physics
本文综述散射振幅与组合几何的深刻联系，梳理从多胞体、拟阵到正构象簇和振幅多面体的结构，阐述正几何如何由几何区域典范确定散射过程的解析函数，并介绍拟阵振幅的最新进展。
Combinatorial flats and Schubert varieties of subspace arrangements
本文通过定义多拟阵的组合平坦格$\mathcal L_P$，解决其缺乏类似拟阵平坦格几何模型的问题，并证明在特征为0的域上可实现时，该格可由子空间配置的Schubert簇建模，推广了已有结果。
Simplicial generation of Chow rings of matroids
本文提出拟阵 Chow 环的单纯生成元表示，借助拟阵商解释建立单项式基与相对 Schubert 拟阵的双射，并证明体积多项式对数凹及 1 次 Hodge–Riemann 关系，且无需 AHK18 的翻转技术。
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