Remark on modular representations of non-commutative algebraic systems

在代数群、量子群和共形场论的研究中出现了几种代数，由于我们可以通过各种方法对这些代数进行详细的分析，所以存在着一个我们可以称之为“特殊非对易代数”的研究领域。在前人对Hecke代数的模表示的研究的基础上，我们将本课题的研究目标定为两个主题：一是将我们的技术发展到新的代数；二是解决Hecke代数的模表示理论中的公开问题。对于第一个目标，我们选取了定义在任意代数闭域上的退化仿射BMW代数，并成功地构造了所有的不可约有限维表示。这是与Mathas和Rui合作的作品。这项工作影响了其他研究者对仿射BMW代数的后续研究。在此期间，Rouquier证明了Cherednik代数…进一步给出了复数域上分圆Hecke代数的拟遗传覆盖。因此，我们有机会对Fock空间进行分类。这给出了一个广泛的视角，推广了首席调查员的分解数定理。通过上述研究进展，我们在项目的最后一年转向了与保形场理论的表象理论联系更紧密的研究，并为下一步的研究项目获得了新的见解。对于第二个目标，首席研究员证明了分圆Hecke代数的模分枝规则，这是研究计划书中提到的预期成果之一。我们还解决了由Dipper，James和Murphy提出的一个猜想，这个猜想已经存在了12年。这是与杰森合作的作品。回想起对称群表示论(和A型Hecke代数)中的Mullineux猜想已经提出多年，最后由Kleshchev(和Brundan)在90年代的S解决了这个猜想，这是一个巨大的成就。将Littelmann路模型应用于Hecke代数的表示理论，得到了著名的Mullineux映射的全新描述。也就是说，即使对于非半单Hecke代数，如果我们工作在路径模型中，Mullineux映射总是由划分的转置给出的。首席研究员发表了13篇论文(其中8篇是参考文献，1篇是参考文献的翻译)，并在研究期间就上述结果和关于经典Hecke代数表示型的结果发表了15次演讲。较少

There are several kinds of algebras that appear in the studies of algebraic groups, quantum groups and conformal field theory As we may carry out detailed analysis of these algebras through applying various kinds of methods, there exists a research field which we might call "Special Noncommutative Algebras". Based on our 'previous research on modular representations of Hecke algebras, we set our goals in this research project in two themes: the first is development of our techniques to new algebras, and the second is to solve open problems in modular representation theory of Hecke algebras. For the first goal, we picked up degenerate affine BMW algebras defined over arbitrary algebraically closed field, and succeeded in constructing all the irreducible finite dimensional representations. This is a joint work with Mathas and Rui. This work influenced several other succeeding research on affine BMW algebras by other researchers. During the period, Rouquier showed that Cherednik algebras … More provide quasihereditary covers of cyclotomic Hecke algebras defined over the field of complex numbers. Thus, we have a chance to categorify Fock spaces. This gives a broad perspective which generalizes the head investigator's decomposition number theorem. By the above mentioned research developments, we have shifted to research which is more closely tied with representation theory of conformal field theory in the last year of the project, and we have obtained new insights for next research project. For the second goal, the head investigator has proved the modular branching rule for cyclotomic Hecke algebras, which was mentioned in the research proposal as one of the expected achievements. We have also settled a conjecture by Dipper, James and Murphy which has been open for 12 years. This is a joint work with Jacon. Recall that the Mullineux conjecture in the representation theory of the symmetric group (and the Hecke algebras of type A) had been open for many years, and it was finally settled by Kleshchev (and Brundan) in 90's, which was a big achievement. By applying Littelmann's path model to representation theory of Hecke algebras, we have obtained completely new description of the famous Mullineux map. Namely, the Mullineux map is always given by transpose of partitions even for non semi-simple Hecke algebras, if we work in the path model. The head investigator has published 13 papers (of which 8 papers are refereed, 1 is translation of a refereed paper) and presented 15 talks on the above results and results on the representation type of Hecke algebras of classical type during the research period. Less