Computing with Hecke algebras

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Hecke algebras are a variety of structures playing a central role in algebra, in particular in the theory of finite groups and their representations. Here, we are interested in two classes of Hecke algebras: Iwahori-Hecke algebras and cyclotomic Hecke algebras, both of which are intimately connected to the representation theory of finite groups of Lie type. Motivated by theoretical questions concerning the structure and representation theory of these algebras, our aim is to develop new computational methods to handle Hecke algebras and their representations. To this end we will make combined use of techniques coming from various branches of computational mathematics, in particular from group theory and commutative algebra. These tools will then be applied to examine substantial interesting, but otherwise inaccessible examples, in order to collect data, to possibly detect previously unknown patterns, and thus to gain structural insights.