Algorithmic methods in the modular representation theory of diagram algebras

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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.