Quasi-isolated blocks and Brauer's height zero conjecture.

准孤立块和布劳尔零高度猜想。

• 批准号: EP/I033637/1
• 负责人: Radha Kessar
• 金额: $ 0.76万
• 依托单位: University of Aberdeen
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI033637%2F1
• 关键词: Quasi; isolated; blocks; Brauer; height

Groups are the mathematical abstraction of symmetry. The realization that unrelated systems and objects can have the same symmetry led mathematicians in the mid-nineteenth century to develop a set of intrinsic axioms for groups which make no reference to the underlying objects whose symmetry they describe. This abstraction defines a group as a collection of objects in which any two objects can be multiplied in a way which follows a prescribed set of rules-the group axioms. This abstract approach had the advantage that any fact discovered for a particular group could then be applied at once to all the contexts in which this group appeared. Even more, it now became possible to formulate questions and prove theorems which held true for all groups.The axioms for groups are short and easy to state, but groups have a rich structure, and their study has been a central theme of mathematics for the past 150 years. One approach to understanding finite groups is through their linear representations. A representation consists of the datum of a group along with a vector space on which the group acts by linear transformations. Since vector spaces have a simple structure compared to groups, the linearisation approach breaks up a complex problem into a collection of simpler problems. This project deals with the modular representation theory of finite groups. In this theory, founded by R. Brauer in the 1930s, one studies simultaneously the representations of finite groups over fields of zero characteristic, such as the complex numbers and over fields of positive characteristic such as the field of integers modulo a prime number. The aim of this project is to complete the classification of l-blocks of finite groups of Lie type and to prove the forward direction of Brauer's height zero conjecture,a problem that has been open for nearly fifty years.

组是对称性的数学抽象。意识到，无关的系统和物体可以具有相同的对称性导致数学家在十九世纪中期为群体开发一组内在公理，这些群体没有提及其描述的对称性的基础对象。该抽象将组定义为一个对象集合，其中任何两个对象都可以以遵循规定的规则集合 - 组公理的方式乘以。这种抽象方法的优势是，对于特定组发现的任何事实都可以立即应用于该组出现的所有情况。更重要的是，现在有可能提出问题并证明对所有小组的定理。小组的公理既短且易于说明，但是小组的结构丰富，他们的研究一直是过去150年来数学的核心主题。理解有限群体的一种方法是通过其线性表示。一个表示由组的基准组成，以及矢量空间，该矢量空间在该空间上通过线性转换作用。由于向量空间与组相比具有简单的结构，因此线性化方法将复杂的问题分解为简单问题的集合。该项目涉及有限群体的模块化表示理论。在这一理论中，由R. Brauer在1930年代创立，同时研究有限基团在零特征的领域（例如复数和积极特征的领域）的表示，例如整数域模拟素数。该项目的目的是完成有限的谎言类型群体的L块分类，并证明Brauer高度零猜想的前进方向，这个问题已经开放了近五十年。