Structure and representation of cyclotomic Hecke algebras

所提出的研究涉及一类重要的有限维代数，即所谓的分圆Hecke代数的结构和表示理论。这些代数是在15年前作为复反射群的群代数的变形而引入的。他们是类似物的岩堀-赫克代数发挥核心作用的代表性理论的有限群的李型。近年来，Iwahori-Hecke代数的许多性质在分圆Hecke代数的理论中有对应。然而，通常需要找到新的方法来证明它们，这与岩堀-赫克代数的方法不同。本项目的目的是继续这一富有成效的方法，通过研究Weyl群的两个最近的结构的类似物。第一个是M. Geck在Coxeter群的情况下对Lusztig代数J进行了研究，这似乎有助于将其推广到分圆Hecke代数。再加上适当修改的概念W-图的明确建设的不可约表示，这应该会导致更好地理解这些代数。第二个出发点是Fiebig的组合概念层的时刻图。

The proposed research concerns the structure and representation theory of an important class of finite dimensional algebras, the so-called cyclotomic Hecke algebras. These algebras were introduced fifteen years ago as deformations of group algebras of complex reflection groups. They are analogues of the Iwahori–Hecke algebras which play a central role in the representation theory of finite groups of Lie type. It has become apparent in recent years that many properties of Iwahori–Hecke algebras have counterparts in the theory of cyclotomic Hecke algebras. Often, though, new approaches have to be found for their proof, different from the one for Iwahori–Hecke algebras. The present project aims at continuing this fruitful approach by investigating analogues of two recent constructions for Weyl groups. The first is the new construction by M. Geck has of Lusztig’s algebra J in the case of Coxeter groups, which seems to lend itself to a generalization to cyclotomic Hecke algebras. Together with a suitably modified concept of W-graph for the explicit construction of irreducible representations this should lead to a better understanding of these algebras. The second starting point is Fiebig’s combinatorial notion of sheaves on moment graphs.