Computing with Hecke algebras

Hecke代数是在代数中扮演核心角色的各种结构，特别是在有限群及其表示理论中。在这里，我们感兴趣的是两类Hecke代数：Iwahori-Hecke代数和环切Hecke代数，它们都与李型有限群的表示理论密切相关。在有关这些代数的结构和表示理论的理论问题的激励下，我们的目标是开发新的计算方法来处理赫克代数及其表示。为此，我们将结合使用来自计算数学各个分支的技术，特别是群论和交换代数。然后，这些工具将被应用于检查大量有趣的，但否则无法访问的示例，以便收集数据，可能检测以前未知的模式，从而获得结构见解。

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.