Computational Methods in Modular Representation Theory

研究者Lux研究了有限域上有限群的计算表示。群可以被认为是对称这一常见概念的数学抽象。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。对象的对称群通常也作用于与给定对象相关的函数空间。这是一个所谓的线性作用，在函数空间是有限维的情况下，群在空间上充当可逆矩阵。这种实现称为群的(矩阵)表示。本文区分了两种情况，一种是矩阵写在特征为0的域(如复数)上，另一种是项在有限域中的情况。这种情况特别有趣，因为这里的表示理论与群的结构紧密相连。研究人员在计算机代数系统GAP中开发了一个共享程序包。这个系统是免费的，有很好的文档，并且得到了广泛的应用。有限特征情形下的表示理论还远未被完全理解。例如，到目前为止，甚至还不是所有有限单群的可约表示都是已知的。Share包实现了一些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群的投影的可分解表示，或者通过由该包提供的数据库来访问它们。当一个人对该群的所有表示感兴趣时，这些表示是非常重要的。从射影不可分解表示可以得到一个称为基本代数的代数，它具有与群相同的表示理论。这种代数比群代数本身小得多，因此在表示式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，但对它几乎一无所知，甚至对特定的组也是如此。共享套餐极大地增加了对特定群体的了解。此外，群的表示与其基本代数的表示之间存在显式的算法联系。Share包包含允许用户使用此连接分析和构建表示的功能。这在群的上同调环或Ext-代数的计算中特别重要。群是一种数学对象，它捕捉到排列和对称的概念--例如，一个正方形在一次旋转90度时的不同方向。群体出现在数学和自然科学中，如物理、化学和生物学。利用团队结构往往会让人对这些领域的问题有更深刻的洞察。研究者开发了表示和计算研究有限群的方法。他在免费提供的软件系统缺口的基础上，在计算机软件中实现了这些方法。他开发的程序包包含计算结果的数据库。公众可通过政府新闻网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，软件包的一个可能应用是在课堂上。它为学生提供免费的表征理论实践体验。

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.