An Adjacency Criterion for Coxeter Matroids

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A _n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.     

The Greedy Algorithm and Coxeter Matroids

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = A (isomorphic to the symmetric group Sym_{_n+1}) and P a maximal parabolic subgroup. The main result of this paper is that for Coxeter matroids, just as for ordinary matroids, the greedy algorithm provides a solution to a naturally associated combinatorial optimization problem. Indeed, in many important cases, Coxeter matroids are characterized by this property. This result generalizes the classical Rado-Edmonds and Gale theorems.A corollary of our theorem is that, for Coxeter matroids L, the greedy algorithm solves the L-assignment problem. Let W be a finite group acting as linear transformations on a Euclidean space , and let

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

准模糊图拟阵基的交换

本文讨论了准模糊图拟阵基的交换定理,在此基础上给出了基有序的准模糊图拟阵的一些性质.     

Constructions of 1 1/2-designs from symplectic geometry over finite fields

In this paper, we construct some 1(1/2)-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the symplectic space and generalized symplectic graphs. Furthermore, these 1(1/2)-designs yield six infinite families of directed strongly regular graphs.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

特征为2的有限域上伪辛群作用在m维全迷向子空间集上的次轨道

本文讨论特征为２的有限域上伪辛几何中，在伪辛群Ｐ＿（ｓ２ｖ＋δ）（Ｆ＿ｑ）作用下ｍ维全迷向子空间可迁集的次轨道．并计算出非平凡轨道的个数及次轨道的长．     

Association schemes based on maximal totally isotropic subspaces in singular pseudo-symplectic spaces

This paper provides some new families of symmetric association schemes based on maximal totally isotropic subspaces in (singular) pseudo-symplectic spaces. All intersection numbers of these schemes are computed.     

On Recognizing Frame and Lifted‐Graphic Matroids

We prove that there is no polynomial with the property that a matroid M can be determined to be either a lifted‐graphic or frame matroid using at most rank evaluations. This resolves two conjectures of Geelen, Gerards, and Whittle (Quasi‐graphic matroids, to appear in J. Graph Theory).     

Basis-Transitive Matroids

We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basis- and point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the structure of two closely related simple matroids, and the set of blocks of imprimitivity bears the structure of a point- and basis-transitive matroid.     

Symplectic Spaces And Ear-Decomposition Of Matroids

Matroids admitting an odd ear-decomposition can be viewed as natural generalizations of factor-critical graphs. We prove that a matroid representable over a field of characteristic 2 admits an odd ear-decomposition if and only if it can be represented by some space on which the induced scalar product is a non-degenerate symplectic form. We also show that, for a matroid representable over a field of characteristic 2, the independent sets whose contraction admits an odd ear-decomposition form the family of feasible sets of a representable Δ-matroid.     

Isotropical linear spaces and valuated Delta-matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n×n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Δ-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.     

Cycles in Circuit Graphs of Matroids

Let G be the circuit graph of any connected matroid. We prove that G is edge-pancyclic if it has at least three vertices. This work is supported by the National Natural Science Foundation(60673047) and the Doctoral Program Foundation of Education Ministry (20040422004) of China.     

辛轨形群胚的辛邻域定理

本文给出一种几何的子轨形群胚的定义,还给出了判定子轨形群胚的依据,并证明了紧子轨形群胚的轨形管状邻域、紧辛子轨形群胚的辛邻域和紧Lagrangian子轨形群胚的Lagrangian邻域的存在性.     

任意基数集上的拟阵之单扩张

对于由Betten和Wenzel于2003年提出的任意基数集上的拟阵其相应的秩公理给予了证明,并将此结果用于研究任意基数集上的拟阵的单扩张问题.     

Self-dual Co des with Symplectic Inner Pro duct

In this paper, we discuss a kind of Hermitian inner product — symplectic inner product, which is different from the original inner product — Euclidean inner product. According to the definition of symplectic inner product, the codes under the symplectic inner product have better properties than those under the general Hermitian inner product. Here we present the necessary and sufficient condition for judging whether a linear code C over Fp with a generator matrix in the standard form is a symplectic self-dual code. In addition, we give a method for constructing a new symplectic self-dual codes over Fp, which is simpler than others.     

On Matroids and Orlik-Solomon Algebras

In this paper we axiomatize combinatorics of arrangements of affine hyperplanes, which is a generalization of matroids, called quasi-matroids. We show that quasi-matroids are equivalent to pointed matroids. On the other hand, the Orlik-Solomon (OS) algebra of a quasimatroid can be constructed. We prove that the OS algebra of a quasi-matroid is isomorphic to the direct image of the OS algebra of a matroid by the linear derivation.AMS Subject Classification: 03B35, 13D03, 52C35.