Algorithmic methods in the modular representation theory of diagram algebras

这个项目的主要目标是在布劳尔代数的研究中发展算法和计算方法。更确切地说，研究对象是有限特征域上的块和分解数，它们与对称群、正交群和辛群的相应问题密切相关。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各种特殊情况的块和分解数。此外，它还将包含许多其他特性，如图的乘法、单元模块的矩阵表示的计算、中心的基等。这种方法的灵感来自于所谓的对称群的juys - murphy元素理论。它们是对称群研究的核心工具，在最近的分类和评分结果中起着重要作用。PI在以前的工作中已经表明，当场的特征与布劳尔代数的程度相比为0或大时，这些元素的类似物也决定了布劳尔代数的分解数。目的是将这一结果细化到更小的特征域，将其扩展到其他相关代数，如BMW-或分区代数，并使用类似的方法确定Brauer代数的块。这也将使我们能够更好地理解juys - murphy元素在Brauer代数的潜在分级和分类结果中的作用。

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.