关于Hall多项式存在性的一个注记

设A是有限域F_q上的有限维代数.著名的Ringel猜测:所有表示有限型的代数A均存在Hall多项式.本文给出了Ringel猜测的一个等价刻画.     

Hall多项式

作为Hall代数的结构常数, Hall数和Hall多项式与对称群的表示和量子群的结构有紧密联系.本文首先引入经典Hall代数和Hall多项式的概念,并阐述其与对称函数的联系.其次,定义有限域上代数的Ringel-Hall代数,并简述其与量子群的关系.最后,本文在Dynkin箭图和仿射箭图情形下,讨论Hall多项式的存在性.     

一类连续匹配问题的Hall型定理

本文研究一类从时间表理论中提出的连续型匹配问题一元素在连续区间内匹配．作为完美匹配存在的充要条件，我们证明了Hall型定理，并说明其在算法上的应用．     

循环单列代数的Hall代数

本文在(Z⁺)^nm上引入序<1和<2,通过(Z⁺)^nm与一个有n个单模且不可分解投射模长度为m的循环单列代数R的有限模同构类的一一对应,它们亦可视为后者的序。由此我们证明了循环单列代数的Hall代数等同于它的Loewy子代数,它的有理扩张有一个BPW型的基,且是一个其相伴分次环为进代斜多项式环的((Z⁺)^nm,<2)滤环。本文结论亦可推广到有限域上的管的Hall代数。     

Hall’s Theorem for Limit Groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Received: May 2006 Revision: May 2007 Accepted: May 2007     

Vanishing integrals for Hall–Littlewood polynomials

It is well known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However, at q = 0 (the Hall–Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers–Szegő polynomials to unify some existing summation type formulas for Hall–Littlewood functions.     

Subgroup Lattices of Hall’s W-Power Groups

We study Hall’s W-power nilpotent groups from the lattice standpoint. A relationship between the structure of a W-power group G and the structure of the lattice of its subgroups L(G) is established.     

The Landau–Hall problem on canal surfaces

In this note we study the Landau–Hall problem on a generalized canal surface and classify all uniform magnetic trajectories of a charged particle moving on such a surface under the action of a uniform magnetic field.     

Hall广义补和群的可解性

研究具有某些特殊性质的广义补,得到了一些可解性的判别条件.如果对G的任意Sylow p-子群P,p∈{2,3}∩丌(G),NG(P)在G中都存在广义补H使H/D是G/D的Hall子群且H/D为幂零群,其中D=(H∩ⅣG(P))G,那么G可解.     

Some remarks on Bridgeland’s Hall algebras

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ? be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→? be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ?. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.