Algorithmic methods in the modular representation theory of diagram algebras

图代数模表示论中的算法方法

• 批准号: 239408760
• 负责人: Dr. Armin Shalile
• 金额: --
• 依托单位: Institut für Algebra und Zahlentheorie (IAZ)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239408760?language=en
• 关键词: Algorithmic; methods; modular; representation; theory

The main goal of this project is to develop algorithmic and computational methods in the study of Brauer algebras. More precisely, the object of study are blocks and decomposition numbers over fields of finite characteristic which are closely related to the corresponding problems for symmetric, orthogonal and symplectic groups. The outcome will be a freely available GAP package which contains an efficient algorithm to compute blocks and decomposition numbers for a wide class of special cases. Furthermore, it will contain many other features such as multiplication of diagrams, computation of matrix representations of cell modules, bases of the center, etc. The approach is inspired by the theory of so called Jucys-Murphy elements for symmetric groups. These are a central tool in the study of symmetric groups and play, for example, a major role in recent results on categorification and grading’s. The PI has shown in previous work that analogues of these elements also determine decomposition numbers of Brauer algebras in the case when the characteristic of the field is either 0 or large compared to the degree of the Brauer algebra. The aim is to refine this result to fields of smaller characteristic, extend it to other related algebras such as the BMW- or partition algebra and determine the blocks of the Brauer algebra by using similar methods. This will also allow us to better understand the role of Jucys-Murphy elements in potential grading and categorification results for the Brauer algebra.

该项目的主要目标是发展Brauer代数研究中的算法和计算方法。更确切地说，研究的对象是块和分解数域的有限特征，这是密切相关的相应问题的对称，正交和辛群。其结果将是一个免费提供的GAP包，其中包含一个有效的算法来计算块和分解数的广泛的一类特殊情况。此外，它将包含许多其他功能，如乘法的图表，计算矩阵表示的细胞模块，基地的中心等的方法是由理论的启发，所谓的Jucys-Murphy元素的对称群。这些是对称群研究中的核心工具，例如，在最近的分类和分级结果中发挥了重要作用。PI在以前的工作中已经表明，这些元素的类似物也决定了Brauer代数的分解数，在这种情况下，该领域的特征是0或大于Brauer代数的程度。其目的是细化这一结果的领域较小的特征，将其扩展到其他相关的代数，如宝马或分区代数和确定块的Brauer代数使用类似的方法。这也将使我们能够更好地理解Jucys-Murphy元素在Brauer代数的潜在分级和分类结果中的作用。