Irreducible modular representations

群的概念在数学中无处不在，并且在科学中有各种各样的应用。作为起点，我们以一个物理对象为例，考虑它的所有对称性：使该对象保持固定的变换（如旋转和反射）。对象的对称群是所有这些对称的列表，以及通过先应用一个然后应用另一个来组合这些对称的操作。这种组合操作满足某些简单的规则-例如，对于每一个对称，都有一个逆对称，它可以让你回到开始的地方。这就引出了抽象群的概念，抽象群是一组符号和一种将这些符号组合起来的方法，这些符号满足相同的规则。群论研究的是存在什么样的抽象群以及它们具有什么样的一般性质。群的研究是数学的一个主要分支，其最令人印象深刻的成就是有限群的分类，这些群是简单的，这意味着它们不能被分解成更小的群。表示论颠倒了上述过程：给定一个抽象群，它问我们如何才能实现这个群作为一个物理对象的对称群。这个问题是通过矩阵表示以代数的方式提出的：n维空间的每个对称性都被编码为n乘n的数字矩阵，组合对称性对应于将矩阵相乘。与对群体进行分类一样，我们希望对简单的（或“不可约”）表示，即那些不能被分解成更小的表示。这个项目着眼于当我们通过取一个素数p并将每个矩阵项替换为模p的余数来改变数制时会发生什么。现在，一个不可约的表示可以变得可约，最重要的任务之一就是找出它的不可约成分是什么。这个项目着眼于这个问题的一个特殊情况，通过要求分类哪些不可约表示在模p约化时仍然是不可约的。这很有趣，因为这意味着分类和构造模p不可约表示的部分任务（通常很难）是自动完成的。在这个项目中，我们将研究单群（以及一些称为准单群和几乎单群的密切相关的群）的这个问题。对于一个重要的几乎单群族，问题得到了解决：它们是对称群S（n），其中n是正整数，S（n）是数字1，.，n.从这个出发点，我们打算解决S（n）的双覆盖的主要问题：这个群是S（n）的两倍大，由带有+或-符号的置换组成，在物理学中有重要的应用。解决了这些群之后，我们打算研究一个非常大的族，即李型群。这些是已经由定义组的矩阵，在各种数字系统和满足各种代数条件。李型有限群的族非常复杂和多样，所以我们打算从族中最简单的成员开始开始。这些是一般线性群，它们是给定数系上所有n乘n矩阵的群，除了有逆矩阵之外没有其他条件。这些群的表示论是很好理解的，并且与对称群的表示论有着非常密切的关系。这意味着它是我们要解决的主要问题的一个很好的初始案例，并作为一般研究Lie型群的测试案例。该项目的成果将是期刊文章和会议演示，以及互联网上的信息库，使数据和初步调查的结果更广泛地可用。

The idea of a group is ubiquitous in mathematics, and has a variety of applications in the sciences. As a starting point, we take a physical object, and consider all its symmetries: transformations (such as rotations and reflections) that leave that object fixed. The symmetry group of the object is the list of all these symmetries together with the operation of combining those symmetries by first applying one then the other. This combining operation satisfies certain simple rules - for example, for every symmetry there is an inverse symmetry which gets you back where you started. This leads to the idea of an abstract group, which is a set of symbols and a way of combining these symbols which satisfies the same rules. Group theory asks what different abstract groups exist and what general properties they have. The study of groups is a major branch of mathematics, and its most impressive achievement is the classification of finite groups which are simple, which means they can't be broken down into smaller groups.Representation theory reverses the above process: given an abstract group, it asks how we can realise that group as the symmetry group of a physical object. This question is asked in an algebraic way through matrix representations: each symmetry of n-dimensional space is encoded as an n by n matrix of numbers, and combining symmetries corresponds to multiplying matrices together. As with classifying groups, we want to classify the simple (or "irreducible") representations of a given group, i.e. those which can't be broken down into smaller representations.This project looks at what happens when we change the number system by taking a prime number p and replacing each matrix entry with its remainder modulo p. Now a representation which was irreducible can become reducible, and one of the most important tasks is to work out what its irreducible constituents are. This project looks at a special case of this question, by asking for a classification of which irreducible representations remain irreducible when reduced modulo p. This is interesting because it means that part of the task of classifying and constructing the irreducible representations modulo p (which in general is very hard) is done automatically. In this project we will look at this question for the simple groups (and some closely related groups called quasi-simple and almost-simple groups). For one important family of almost simple groups the problem is solved: these are the symmetric groups S(n), where n is a positive integer and S(n) is the group of all permutations of the numbers 1,...,n. From this starting point, we intend to solve our main problem for the double cover of S(n): this group is twice as large as S(n), consisting of permutations accompanied by a + or - sign, and has important applications in physics.Having tackled these groups, we intend to look at a very large family, namely the groups of Lie type. These are groups which are already by definition groups of matrices, over various number systems and satisfying various algebraic conditions. The family of finite groups of Lie type is very complex and varied, so we intend to start with the easiest members of the family to begin with. These are the general linear groups, which are the groups of all n by n matrices over a given number system, with no additional condition other than having inverses. The representation theory of these groups is quite well understood, and has a remarkably close relationship to the representation theory of the symmetric groups. This means that it is an excellent initial case of our main question to solve, and acts as a test case for studying the groups of Lie type in general.The outcomes of the project will be journal articles and conference presentations, together with a repository of information on the internet making data and the outcomes of preliminary investigations more widely available.