旗流形的上同调环的自同态

本文应用李群理论,对紧单李群Ｇ,给出了旗流形 Ｇ／Ｔ的上同调环自同态的完全分类,并对典型群计算了相应自同态的Ｌｅｆｓｃｈｅｔｚ数．     

Closure ordering and the Kostant-Sekiguchi correspondence

Let be a real semisimple Lie group with Lie algebra . The Kostant-Sekiguchi correspondence is a bijection between nilpotent orbits on and nilpotent orbits on . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant-Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

     

On orbit closures of spherical subgroups in flag varieties

Let be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on with finitely many orbits, and let V be an H-orbit closure in . Expanding the cohomology class of V in the basis of Schubert classes defines a union V₀ of Schubert varieties in with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V₀ is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V₀ in . Thus, for any effective line bundle L on , the restriction map is surjective, and for all . Received: April 17, 2000     

A note on the Glauberman correspondence

In this paper, we will investigate the Glauberman-Watanabe correspondence between the characters and blocks. Motivated by the work of Puig and Zhou [10 Puig, L., Zhou, Y. (2012). Glauberman correspondents and extensions of nilpotent block algebras. J. London Math. Soc. (2) 85809–837.[Crossref] , [Google Scholar]] which is about the extension of a nilpotent block and its Glauberman correspondent, we will modify the basic Morita equivalence obtained in their work and get a new equivalence such that the correspondence between characters induced by this equivalence coincides with the Glauberman correspondence under the situation that the block is a p-extension of a nilpotent block.     

Recognizing Schubert Cells

We address the problem of distinguishing between different Schubert cells using vanishing patterns of generalized Plücker coordinates.     

Almost Linear Nash Groups

A Nash group is said to be almost linear if it has a Nash representation with a finite kernel. Structures and basic properties of these groups are studied.     

On Jordan quadruple systems with quadripotents

The purpose of this article is to show the link between Jordan quadruple systems with quadripotents and Jordan algebras. We also extend the notions of the orthogonality, primitivity, and minimality of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We refine the Peirce decomposition of a Jordan quadruple system with respect to a quadripotent to be with respect to a system of orthogonal quadripotents and get the multiplication rules of the Peirce spaces. We show that the notions of primitive and minimal quadripotents coincide in a Jordan quadruple system.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

A note on the fundamental group of Finsler manifolds with nonnegative flag curvature

In this note we prove that the fundamental group of any forward complete Finsler manifold with nonnegative flag curvature is finitely generated provided the line integral of T-curvature is small. In particular, the fundamental group of any forward complete Berwald manifold with nonnegative flag curvature is finitely generated.     

Bruhat order, smooth Schubert varieties, and hyperplane arrangements

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincaré polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.     

On computing the integral closure

In Hai and Thin [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar]], there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring D is central (see [1 Hai , B. X. , Thin , N. V. On locally nilpotent subgroups of GL ₁(D). Communications in Algebra 37 ( 2 ): 712 – 718 . [Google Scholar], Theoerem 2.2]). Unfortunately, there is some mistake in the proof of this theorem. In this note, we give the another proof of this theorem.     

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach that avoids geometry, this paper studies the structure of K_T(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis (the double Grothendieck polynomials), where is the structure sheaf of the Schubert variety X_w. For rank two cases we compute the corresponding structure constants of the ring K_T(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for H_T^*(G/B)). Let [X^λ]K_T(G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove “Pieri–Chevalley formulas” for the products , , , and , where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), H_T^*(G/B) and H^*(G/B).