Xun Dong
Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.
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In an unpublished manuscript of 1992, Johan Karlander has given an axiomatization of affine oriented matroids. A closer examination of the text revealed an invalid construction and an incorrect argument in the proof of his main theorem. This paper provides an alternative argument to fix and slightly simplify the proof of the main theorem.
Matroids Arising From Nested Sequences of Flats In Projective And Affine Geometries
本文研究了由射影与仿射几何中平坦子集嵌套序列导出的拟阵，将Nelson和Nomoto关于二元目标禁止诱导限制的结果推广至$GF(q)$上的射影几何，并确定了仿射目标的禁止诱导限制。
Almost Affine Vector Rank-Metric Codes
本文定义几乎仿射向量秩度量码，证明其自然诱导q-拟阵，并建立穿孔和缩短操作与q-拟阵的限制和收缩的对应关系，同时证明秩权重和形式对偶距离分布由诱导的q-拟阵决定，最后给出严格几乎仿射秩度量码的构造实例。
An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got beautiful results. Developing their theory, we will show the following. (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex (a regular CW complex associated with it) of the corresponding affine oriented matroid is a contractible homology manifold with boundary. This is closely related to Dong's theorem, which used to be "Zaslavsky's c…
We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such matroids. We give algorithms to find the hyperplanes and the hamiltonian circuits in such cases.
Finitary affine oriented matroids
本文开创仿射定向拟阵的公理化研究，建立子式、重定向等基本概念，证明有限情形下的可剥壳性，并利用群作用与多重Tutte多项式推广环面排列的计数性质。
We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.
We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of Montellano-Ballesteros and Strausz about crabbed connectivity of cocircuit graphs of uniform oriented matroids.
Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid is given by the point sets in general position of an affine space, leading to the concept of an affine design. Accordingly, a t-(n, k, $λ$) affine design of order q is a collection B of (k-1)-dimensional spaces in the affine geometry A = AG(n-1, q) such that each (t-1)-dimensional space in A is contained in exactly $λ$ spaces of B. In the case $λ$ = 1, as u…
Counter Example to a Strong Matroid Minor Conjecture
本文通过在仿射无限 Grassmannian 的坐标环中显式构造 $S_\infty$-稳定理想的严格升链，否定了 Draisma 的强形式拟阵小猜想，同时指出其拓扑 Noether 性猜想仍开放。
Oriented matroid structures on rank 3 root systems
本文证明了秩3仿射根系在满足Weyl群等变且限制为秩2子系统通常拟阵结构的条件下，具有唯一的定向拟阵结构，并将该唯一性结论推广至有限根系及“干净”秩3根系。
An Intersection Matrix for Affine Hyperplane Arrangements
本文为实仿射超平面排列定义整数相交矩阵及其 $q$-形变，证明其行列式仅依赖于底层拟阵的组合结构，并给出表示论解释。
Unlabeled sample compression schemes for oriented matroids
本研究针对样本压缩猜想，利用有向拟阵的度量与组合结构，构建了以VC维为界的有向拟阵拓扑位类的无标记样本压缩方案，结果可推广至仿射及具角剥离性质的有向拟阵复形的拓扑位，主要工具为特定有向拟阵规划的解。
The tropical critical points of an affine matroid
本文证明仿射拟阵的热带临界点数等于其 beta 不变量，并给出等价计数条件，回答了 Sturmfels 的问题。
Unavoidable flats in matroids representable over prime fields
本文证明高阶简单 GF(p)-可表示拟阵必含独立、射影或仿射几何型的秩-k flat，并由此推出二染色下的单色 flat Ramsey 型定理。
Las Vergnas Cube Conjecture states that the cube matroid has exactly one class of orientations. We prove that this conjecture is equivalent to saying that the oriented matroid of the affine dependencies of the n-cube can be reconstructed from the underlying matroid and one of the following partial lists of signed circuits or cocircuits: A) the signed circuits of rank 3, or B) the positive signed cocircuits.
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recove…
In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type $A$ for rectangular type semisimple nil-element $N$ and calculate the relative position between irreducible components. In particular, we use the affine matrix ball construction to show the relative position map is compatible with the Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and van Leeuwen.
This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter group in real vector spaces. We settle the question for finite Coxeter groups, irreducible affine Weyl groups and all rank three Coxeter groups. In these cases, the oriented matroid structure is unique unless $W$ is of type $\widetilde{A}_n, n\geq 3$, in which case there are three possibilities.
In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular …
Affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an affine generalization of Robinson-Schensted correspondence. We show that AMBC gives a simple way to compute a distinguished (or Duflo) involution in each Kazhdan-Lusztig cell of affine symmetric groups. We then use AMBC to give the first known canonical presentation for the asymptotic Hecke algebras of extended affine symmetric groups. As an application, we show that AMBC gives a conceptual way to compute Lusztig-Vogan bijection. For the latter we build upon prior works of Achar a…
In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, ρ)$ where $P$ and $Q$ are tabloids and $ρ$ is a dominant weight. The weights $ρ$ get …
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Furthermore, we obtain polynomial time recognition a…
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe …
Chow polynomials of rank-uniform labeled posets
研究秩均匀标记偏序集的Chow多项式实根性质，揭示其与matroid代数几何结构的关联。
Gårding Polynomials
本文引入了Gårding多项式，这是一种实多变量多项式，其正区域在正向量平移下保持不变且在严格正的仿射变换下封闭。研究证明了这种几何描述等价于通过极化转换到多仿射设置以及基于偏导数的递归准则。Gårding多项式严格扩展了实稳定多项式类，同时保留了许多结构性质。此外，Gårding多项式在多种组合生成函数下保持Gårding性质，应用于生成函数的负依赖结果，以及某些矩阵类的特征多项式。
Labeled sample compression schemes for complexes of oriented matroids
本文证明 VC 维为 d 的定向拟阵复形的 tope 集合存在大小为 d 的恰当有标号样本压缩方案，推广了多类已有结果，并借助组合结构与偏立方体理论推进样本压缩猜想。
This is the second part of a two-part investigation. We continue the study of a class of balanced urn schemes on balls of two colors (white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn and ball addition rules are applied; the special case $m=1$ of sampling only a single ball coincides with ordinary balanced urn models. We consider these multiple drawings under sampling with or without replacement. For the class of affine conditional expected value, we study the number of white balls after $n$ steps. The affine class is parametrized by $Λ$, specified by the ratio…
This is a research endeavor in two parts. We study a class of balanced urn schemes on balls of two colours (say white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn, and ball addition rules are applied. We consider these multiple drawings under sampling with or without replacement. We further classify ball addition matrices according to the structure of the expected value into affine and nonaffine classes. We give a necessary and sufficient condition for a scheme to be in the affine subclass. For the affine subclass, we get explicit results for the expected value …
For each odd integer $k\ge 5$, we prove that, if $M$ is a simple rank-$r$ binary matroid with no odd circuit of length less than $k$ and with $|M| > k 2^{r-k+1}$, then $M$ is isomorphic to a restriction of the rank-$r$ binary affine geometry; this bound is tight for all $r\ge k-1$. We use this to give a simpler proof of the following result of Govaerts and Storme: for each integer $n\ge 2$, if $M$ is a simple rank-$r$ binary matroid with no $PG(n-1,2)$-restriction and with $|M| > \left(1-\frac{11}{2^{n+2}}\right) 2^r$, then $M$ has critical number at most $n-1$. That result is a geometric anal…
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 affine Tverberg theorem revisited
本文证明Bárány--Kalai--Tverberg猜想，将仿射Tverberg定理从单纯形推广至任意凸多胞体与单纯球，确保存在r个不相交面使仿射像有公共交点。
Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects
本文研究 Lorentzian 多项式空间的拓扑，证明其与三角超域上的细 Schubert 胞腔同胚，并引入 Hausdorff 紧化，连接代数几何中的经典构造。
Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumera…
A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological hyperplanes in H, if their intersection is nonempty, meet in a subspace that is a topological hyperplane in the intersection of any k-1 of them; but two topological hyperplanes that do intersect need not cross each other. If every intersecting pair does cross, the arrangement is affine. The number of regions formed by an arrangement of topological hyperplanes has the …
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality, twisted homology groups, monodromy theory and multidimensional hypergeometric functions. The emphasis on the most geometrical explanation is done; applications and analogies in the differential topology are outlined.
In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0, 1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is uncovered? The $k=1$ case is the well-known Alon-Füredi theorem which says a minimum of $n$ affine hyperplanes is required, proved by the Combinatorial Nullstellensatz. We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integr…
Volume growth of Funk geometry and the flags of polytopes
本文研究Funk几何中球的体积增长，提出并部分验证了与Mahler和旗猜想相关的极小化猜想，并计算了多胞形情形的高阶渐近项，推广了Santaló点概念。
Soliton cellular automata for the affine general linear Lie superalgebra
本文基于 Kwon–Okado 的 Kirillov–Reshetikhin 晶体，将 Hikami–Inoue 面向 $\mathfrak{gl}(m|n)$ 的盒球系统推广至列表象，并验证其在特定条件下保持孤子行为。
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