On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for G = SL(2) × C~×and(b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.     

On Sylow Subgroups of Abelian Affine Difference Sets

An n-subsetD of a group G of order is called an affine difference set of G relativeto a normal subgroup N of G of order if the list of differences containseach element of G-N exactly once and no elementof N. It is a well-known conjecture that if Dis an affine difference set in an abelian group G,then for every prime p, the Sylow p-subgroupof G is cyclic. In Arasu and Pott [1], it was shownthat the above conjecture is true when . In thispaper we give some conditions under which the Sylow p-subgroupof G is cyclic.     

On Spencer Cohomologies of Affine Algebraic Varieties

This paper contains proofs and examples for some key facts about algebraic Spencer cohomologies preannounced elsewhere. The author hopes that here he has managed to build a more systematic discourse of the issues raised in earlier works.     

On Existence Varieties of E-solid Semigroups

Within the class of regular E-solid semigroups, a theory of e-varieties including appropriate notions of biidentities and biinvariant congruences is presented, such that, together with bifree objects, these notions inherit the properties and interrelations well known from universial algebra. This theory generalizes the previously developed such theory for orthodox semigroups. As an application, the bifree objects in certain e-varieties of E-solid locally orthodox semigroups, which are constructed by means of Malcev products from a varities of bands, groups and completely simple semigroups, are described as subsemigroups in suitable Pastijn products of some bands by relatively bifree completely simple semigroups. As a consequence, it follows that every regular E-solid locally orthodox semigroup regularly divides a so-called solid Pastijn product of a band by a completely simple semigroup.     

On the Varieties of Parabolic Subgroups, their Generalizations and Combinatorial Applications

Investigations of homogeneous varieties T=(G:P) of all cosets of finite Coxeter or Chevalley groups G by their maximal parabolic subgroups P had been conducted at the Kalunin seminar at Kiev State University since the 1970s, as were investigations of their corresponding permutation groups, geometries and association schemes.In I. A. Faradev et al. (eds), Investigations in Algebraic Theory of Combinatorial Objects (Kluwer Acad. Publ., 1994), one can find some results on the investigation of noncomplete Galois correspondence between fusion schemes of the orbital scheme for (G,T) and overgroups of (G,T), as well as calculations of the intersectional indices of the Hecke algebra of (G,T). We will discuss additional results on this topic and consider questions related to the following problems: embeddings of varieties (G:P) into the Lie algebra corresponding to Chevalley group G; interpretations of Lie geometries, small Schubert cells, connections between the geometry of G and its associated Weyl geometry in terms of linear algebra, and applications of these problems to calculations performed in Lie geometries and association schemes; constructions of geometric objects arising from Kac–Moody Lie algebras and superalgebras, and applications of these constructions to investigations of graphs of large girth and large size.     

On the Existence of $ G $ -Permutable Subgroups in Simple Sporadic Groups

Siberian Mathematical Journal - A subgroup $ A $ of a group $ G $ is $ G $ -permutable in  $ G $ if for every subgroup $ Bleq G $ there is $ xin G $ satisfying $ AB^{x}=B^{x}A $ ....     

On Representations of Affine Hecke Algebras of Type B

Ariki’s and Grojnowski’s approach to the representation theory of affine Hecke algebras of type A is applied to type B with unequal parameters to obtain – under certain restrictions on the eigenvalues of the lattice operators – analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type A. Research supported by the Studienstiftung des deutschen Volkes.     

A Note on the Canonical Divisor of Quasi-homogeneous Affine Algebraic Varieties

We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.     

幂零Hall π-子群的存在性

本文首先将Ｈａｌ定理推广为：设Ｎ为Ｇ的正规子群，若Ｎ为Ｅｎπ群，Ｇ／Ｎ为Ｄπ群，则Ｇ为Ｄπ群．在此基础上得到了群Ｇ为Ｅｎπ群的充要条件为：（１）Ｇ存在正规子群Ｎ，满足Ｎ及Ｇ／Ｎ为Ｅｎπ群；（２）对任意ｐ∈π，任意ｑ∈π ｛ｐ｝及任意ｐ 元素ｘ，ＣＧ（ｘ）含Ｇ的Ｓｙｌｏｗｑ 子群．另外，我们对非Ａｂｌｅ单群的情形也进行了一些讨论．     

p-Fitting子群的CAP-嵌入子群

群G的子群H称为G的CAP-嵌入子群,如果对于|H|的每个素因子p,存在G的某个CAP-子群K,使得H的某个Sylow p-子群也是K的一个Sylowp-子群.本文通过假定G的p-Fitting子群F_p(G)的某个Sylow p-子群的每个极大子群是G的CAP-嵌入子群,得到一些新的结果.     

Gauge Modules for the Lie Algebras of Vector Fields on Affine Varieties

Algebras and Representation Theory - For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $mathcal {A}$ of functions...     

On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993 Helminck , A. G. , Wang , S. P. ( 1993 ). On rationality properties of involutions of reductive groups . Adv. Math. 99 ( 1 ): 26 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.     

On the Affine Temperley-Lieb Algebras

The paper describes the cell structure of the affine Temperley–Liebalgebra with respect to a monomial basis. A diagram calculusis constructed for this algebra.     

The Growth Rate of Symplectic Homology and Affine Varieties

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.     

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Let G be a connected reductive algebraic group over ?, and let Λ _G ⁺ be the monoid of dominant weights of G. We construct integrable crystals B^G(λ), λ ∈ Λ _G ⁺ , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps \(P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} \) in terms of multiplication in generalized transversal slices. Let L ? G be a Levi subgroup of G. We describe the functor Res _L ^G : Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.     

Sanya Lectures: Geometry of Spherical Varieties

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.