Resonant Local Systems on Complements of Discriminantal Arrangements and Sl2Representations

We calculate the skew-symmetric cohomology of the complement of a discriminantal hyperplane arrangement with coefficients in local systems arising in the context of the representation theory of the Lie algebra . For a discriminantal arrangement in ^k, the skew-symmetric cohomology is nontrivial in dimension k–1 precisely when the 'master function' which defines the local system on the complement has nonisolated criticalpoints. In symmetric coordinates, the critical set is a union of lines. Generically, the dimension of this nontrivial skew-symmetric cohomology group is equal to the number of critical lines.     

Basic Derivations for Subarrangements of Coxeter Arrangements

We prove that various subarrangements of Coxeter hyperplane arrangements are free. We do this by exhibiting a basis for the corresponding module of derivations. Our method uses a theorem of Saito [24] and Terao [30] which checks for a basis using certain divisibility and determinantal criteria. As a corollary, we find the roots of the characteristic polynomials for these arrangements, since they are just one more than the degrees in any basis of the module. We will also see some interesting applications of symmetric and supersymmetric functions along the way.     

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the Riemann hypothesis if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system A _{n – 1}. The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.     

Gauss–Man i n Conne ction s for Arrangements, I E igenvalues

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

对称多胞形的极值性质及其应用

凸多胞形现代理论的主要成就是被称之为Dehn-Sommerville关系的上界定理和下界定理,它们属于凸多胞形的经典组合理论．本文建立了关于对称凸多胞形的两个极值定理,它们可视为凸多胞形度量理论中的上界定理和下界定理,另外给出了两个极值定理的一个应用．     

Convex and Abstract Polytopes (Thursday, May 19, 2005 to Sunday, May 22, 2005)

Summary A report on the Convex and Absract Polytopes Workshops held at the Banff International Research Station and The University of Calgary, May 19--22, 2005.     

Desargues Theorem, Dynamics, and Hyperplane Arrangements

The Desargues theorem is a basic theorem in classical projective geometry. In this paper we generalize Desargues theorem in the direction of dynamical systems. Our result comprises an infinite family of configurations, having unbounded complexity. The proof of the result involves constructing special kinds of hyperplane arrangements and then projecting subsets of them into the plane.     

An Atlas of Small Regular Abstract Polytopes

Summary This article announces the creation of an atlas of small regular abstract polytopes. The atlas contains information about all regular abstract polytopes whose automorphism group has order 2000 or less, except those of order <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>512k$ where $k\geq1$. The article explains also the techniques used to create the atlas, and gives some summary tables. At the time of printing, the url for the atlas is http://<a href = "http://www.abstract-polytopes.com/atlas">www.abstract-polytopes.com/atlas</a>.     

Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes. Received January 24, 1997, and in revised form April 8, 1997.     

关于多胞形一个新仿射不变量的应用

用组合极值方法导出了n维欧氏空间中关于原点对称的一个凸多胞形子类上一个新的仿射不变量(最近由Lutwak,Yang和Zhang引入)的解析表达式,并给出了其在凸多胞形Minkowski问题的一个应用．     

Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

A Theory of Nets for Polyhedra and Polytopes Related to Incidence Geometries

Our purpose is to elaborate a theory of planar nets or unfoldings for polyhedra, its generalization and extension to polytopes and to combinatorial polytopes, in terms of morphisms of geometries and the adjacency graph of facets.     

Arrangements and Ranking Patterns

In the unidimensional unfolding model, given m objects in general position on the real line, there arise 1 + m(m − 1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. Change of the position of these m objects results in change of the ranking pattern. In this paper we use arrangement theory to determine the number of ranking patterns theoretically for all m and numerically for m ≤ 8. We also consider the probability of the occurrence of each ranking pattern when the objects are randomly chosen. Received March 5, 2005     

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

Hanlon and Stanley's Conjecture and the Milnor Fibre of a Braid Arrangement

Let A be a real arrangement of hyperplanes. Let B = B(q) be Varchenko's quantum bilinear form of A, introduced [15], specialized so that all hyperplanes have weight q. B(q) is nonsingular for all complex q except certain roots of unity. Here, we examine the kernel of B at roots of unity in relation to the topology of the hyperplane singularity.We use Varchenko's work [16] to relate B(q) to a Salvetti complex for the Milnor fibration of A. This paper's main result is specific to the arrangement of reflecting hyperplanes associated with the A _{n – 1} root system. We use a geometric property of the Milnor fibre to resolve a conjecture due to Hanlon and Stanley regarding the -module structure of the kernel of B(q) at certain roots of unity.     

L\"obell多面体上的小覆盖

计算了L\"{o}bell多面体上的小覆盖的等变微分同胚类的个数. 在1991年, Davis和Januszkiewicz提出了小覆盖的概念, 给出了组合和拓扑间的一种直接联系, 并证明了单凸多面体上的特征映射($\mathbb{Z}_2^n$染色)与该多面体上的小覆盖一一对应. 文中作者给出了L\"{o}bell多面体上的自同构群和染色规律, 结合Burnside引理计算了一般的L\"{o}bell多面体上的小覆盖的等变微分同胚类的个数.