D_l和E_6型Weyl群的扭子群的扩群

本文决定了 Dl 和 E6 型 Weyl群扭子群的所有扩群 ,这为确定相应 Chevalley群扭子群的所有扩群奠定了基础 .     

有限域上F4和G2型Chevalley群的二元生成

本文证明了有限域上F4和G2型Chevalley群均可由两个元素生成.     

Chevalley群的一类子群的研究

设G＝L（F）是特征不为2的域F上Chevalley群,型为B1（l≥4）,Cl（l≥3）,Dl（l≥5）,E6,E7,E8或F4之一．当L（F）型为B4或F4时还假设F=F2设Lα1是L（F）的一类Levy子群．本文决定Lα1的正规化子在L（F）中的极大性．     

Levi子群Lα（n（α）=1）在Chevalley群G（Φ，F）中的扩群

设L是复数域上单李代数，具有不可约根系Φ，固定基п。设F是一个特征不为2的域，且不是三元域，G（Φ，F）是F上Φ型的Chevalley群。设α∈п，Φα表示Φ的一种类型子根系。当n(α)＝1，且Φ是Bl(l≥3),Dl(l≥4),E6，E7或E8之一时，本文决定了Levi子群Lα在G（Φ，F）中的所有扩群。     

有限域上Bl型Chevalley群的二元生成

证明了有限域上Bl型Chevalley群可由两个元素生成     

特征不为 2 的有限域上酉群的极小生成元集

设K=Fq2为含有q2个元素的有限域,q为奇素数的幂,*：a→a*=aq是Fq2的一个二阶自同构．本文用几何方法证明了除K为F32而n=4的情形外,Fq2上的酉群Un（V）可由2个元素生成．     

Ixr(a)的有限型Ixor(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型IC or (a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数IXr(a)的Weyl群W同构于有限型IXr(a)的未定Weyl群.     

IX_r(a)的有限型IX_r~°(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型I(?)r(a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数,IXr(a)的Weyl群W同构于有限型I(?)r(a)的未定Weyl群.     

仿射Weyl群上的Demazur乘积

Demazure乘积是定义在一般Coxeter群上的一类幺半群乘积.它自然地出现在李理论中的不同领域中.本文将研究仿射Weyl群上Demazur乘积.我们的主要结果是发现了它与有限Weyl群上的量子Bruhat图之间的一个紧密联系.作为应用,我们给出了仿射Weyl群最低双边胞腔元素之间Demazure乘积的显示表达式,并得到了最低双边胞腔元素的一般牛顿点以及Lusztig-Vogan映射的具体刻画.     

Levi子群Lα(n(α)=1)在Chevalley群G(Ф,F)中的扩群

设L是复数域上单李代数,具有不可约根系Ф,固定基П.设F是一个特征不为2的域,且不是三元域,G(Ф,F)是F上Ф型的Chevalley群.设α∈П,Фα表示Ф的一种类型子根系.当n(α)=1,且Ф是Bl(l≥3),Dl(l ≥ 4),E6,E7,或E8之一时,本文决定了Levi子群Lα在G(Ф,F)中的所有扩群.     

On quotients of Coxeter groups under the weak order

We consider quotients of finitely generated Coxeter groups under the weak order. Björner and Wachs proved that every such quotient is a meet semi-lattice, and in the finite case is a lattice [Björner and Wachs, Trans. Amer. Math. Soc. 308 (1988) 1–37]. Our result is that the quotient of an affine Weyl group by the corresponding finite Weyl group is a lattice, and that up to isomorphism, these are the only quotients of infinite Coxeter groups that are lattices. In this paper, we restrict our attention to the non-affine case; the affine case appears in [Waugh, Order 16 (1999) 77–87]. We reduce to the hyperbolic case by an argument using induced subgraphs of Coxeter graphs. Within each quotient, we produce a set of elements with no common upper bound, generated by a Maple program. The number of cases is reduced because the sets satisfy the following conjecture: if a set of elements does not have an upper bound in a particular Coxeter group, then it does not have an upper bound in any Coxeter group whose graph can be obtained from the graph of the original group by increasing edge weights.     

Reflection functors for quiver varieties and Weyl group actions

We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowys construction of Springer representations of the Weyl group. Mathematics Subject Classification (2000):Primary 53C26; Secondary 14D21, 16G20, 20F55, 33D80Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan.     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter-example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski–Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.     

Reflection groups on Riemannian manifolds

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds. Mathematics Subject Classification Primary 51F15, 53C20, 20F55, 22E40     

Extended symmetric spaces and θ-twisted involution graphs

Abstract

For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for G = S_n, θ an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and θ.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

A linear interpretation of the flag geometries of Chevalley groups

It is proven that the flag geometry of a Chevalley group can be derived from the flag geometry of its Weyl group by using a linear covering defined by the author. To prove this, the author regards elements of the Weyl group geometry as vectors of a Euclidean space in such a way that the incidence of vectors is defined by their scalar products.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 383–387, March, 1990.     

Quasi-Minuscule Quotients and Reduced Words for Reflections

We study the reduced expressions for reflections in Coxeter groups, with particular emphasis on finite Weyl groups. For example, the number of reduced expressions for any reflection can be expressed as the sum of the squares of the number of reduced expressions for certain elements naturally associated to the reflection. In the case of the longest reflection in a Weyl group, we use a theorem of Dale Peterson to provide an explicit formula for the number of reduced expressions. We also show that the reduced expressions for any Weyl group reflection are in bijection with the linear extensions of a natural partial ordering of a subset of the positive roots or co-roots.