Computational Methods in Modular Representation Theory

研究者研究有限域上有限群的计算表示。群可以被认为是对一般对称概念的数学抽象。因此，它们出现在数学和自然科学中，例如物理、化学和生物学。一个对象的对称群通常也作用于与给定对象相关的函数空间。这就是所谓的线性作用在函数空间是有限维的情况下群是空间上的可逆矩阵。这样的实现称为群的（矩阵）表示。在这种情况下，可以区分两种情况，一种是矩阵写在特征为0的域上，比如复数，或者是条目在有限域中。这个案例特别有趣，因为在这里再现理论与群体结构紧密相连。研究者在计算机代数系统GAP中开发了一个共享程序包。这个系统是免费的，有完整的文档，并且被广泛使用。有限特征情况下的表示理论还远远没有被完全理解。例如，到目前为止，甚至不是所有有限单群的可约表示都是已知的。共享包实现算法来帮助解决有关有限群表示的几个问题。首先，它使用户能够计算给定组的投影不可分解表示，或者通过包提供的数据库访问它们。当一个人对群体的所有表征有兴趣时，这些表征是至关重要的。从这些射影不可分解的表示中，我们推导出一种称为基本代数的代数，它与群具有相同的表示理论。这个代数比群代数本身小得多，因此在表示计算中也有重要的应用。因此，共享包还包含计算基本代数的函数。尽管基本代数是一个非常重要的不变量，但我们对它几乎一无所知，即使是对特定的群。共享包极大地增加了对特定群体的了解。此外，在群的表示与其基本代数的表示之间存在显式的算法连接。共享包包含允许用户使用此连接分析和构造表示的函数。这在计算上同环或群的外代数时特别重要。群是一个数学对象，它捕捉了排列和对称的概念——例如，一个正方形每次转动90度时的不同方向构成了一个群。群体出现在数学和自然科学领域，如物理、化学和生物学。利用群体结构通常会导致对这些领域问题的深刻见解。研究者开发了表示和计算研究有限群的方法。他在计算机软件中实现了这些方法，建立在免费的软件系统GAP上。他开发的包包含计算结果的数据库。公众可通过gap系统或在万维网上查阅这些资料。GAP本身是在教育环境中使用的，作为GAP的一部分，软件包的一个可能的应用是在课堂上。它为学生免费提供表征理论的实践经验。

Lux The investigator studies computational representations of afinite group over finite fields. Groups can be thought of as themathematical abstraction of the common notion of symmetries. Assuch they arise in mathematics and natural sciences such as forexample physics, chemistry, and biology. The symmetry group of anobject usually also acts on function spaces that are related tothe given object. This is a so-called linear action and in thecase where the function space is finite dimensional the groupacts as invertible matrices on the space. Such a realization iscalled a (matrix-) representation of the group. In this contextone distinguishes two cases, the one where the matrices arewritten over a field of characteristic 0 such as the complexnumbers, or the case where the entries are in a finite field.This case is particularly interesting since here therepresentation theory is tightly connected to the structure ofthe group. The investigator develops a share package of programsin the computer algebra system GAP. This system is free ofcharge, well documented, and widely used. The theory ofrepresentations in the finite characteristic case is far frombeing fully understood. For example, up to now not even all theirreducible representations for all finite simple groups areknown. The share package implements algorithms to help solveseveral questions concerning the representations of a finitegroup. First of all it enables the user to compute the projectiveindecomposable representations of a given group or alternativelyaccess them via a data base that is provided by the package.These representations are of fundamental importance when one isinterested in getting an overview of all representations of thegroup. From the projective indecomposable representations one canderive an algebra called the basic algebra that has the samerepresentation theory as the group. This algebra is much smallerthan the group algebra itself and hence also has importantapplications in computing with representations. The share packagetherefore also contains functions to compute the basic algebra.Even though the basic algebra is a very important invariant,barely anything is known about it even for specific groups. Theshare package increases the knowledge about specific groupstremendously. Moreover, there is an explicit algorithmicconnection between representations of a group and that of itsbasic algebra. The share package incorporates functions thatallow the user to analyze and construct representations usingthis connection. This is of particular importance in computationsof the cohomology ring or the Ext-algebra of a group. A group is a mathematical object that captures notions ofarrangement and symmetry -- for example, the differentorientations of a square when turned ninety degrees at a timecomprise a group. Groups arise in mathematics and naturalsciences such as physics, chemistry, and biology. Exploiting thegroup structure often leads to deep insights into problems inthese areas. The investigator develops ways to represent andcomputationally study finite groups. He implements these methodsin computer software, building on the freely available softwaresystem GAP. The package he develops contains data bases ofcomputed results. These are accessible to the public via the GAPsystem and alternatively on the world wide web. GAP itself isused in the educational environment and as part of GAP onepossible application of the software package is in the classroom. It gives students hands-on experience with representationtheory free of charge.