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，由于我们可以通过应用各种方法对这些代数进行详细的分析，因此存在一个研究领域，我们可以称之为“特殊非交换代数”。基于我们以前对Hecke代数模表示的研究，我们将本课题的研究目标分为两个方面：一是将我们的方法发展到新的代数上，二是解决Hecke代数模表示理论中的一些公开问题。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不可约的有限维表示。这是与Mathas和Rui的合作作品。这项工作影响了其他几个成功的研究仿射宝马代数的其他研究人员。在此期间，Rouquier证明了Cherednik代数 ...更多信息 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，推广了首席研究员的分解数定理。通过上述研究进展，我们在项目的最后一年转向了与共形场论的表示理论联系更紧密的研究，并为下一个研究项目获得了新的见解。对于第二个目标，首席研究员已经证明了分圆Hecke代数的模分支规则，这在研究计划中作为预期成果之一提到。我们还解决了迪普、詹姆斯和墨菲提出的一个猜想，这个猜想已经公开了12年。这是与Jacon的合作。回想一下，对称群（以及A型Hecke代数）的表示论中的Mullineux猜想已经公开了很多年，最终由Kleshchev（和Brundan）在90年代解决，这是一个巨大的成就。将Littelmann的路模型应用于Hecke代数的表示论，得到了著名的Mullineux映射的全新描述。也就是说，在路径模型下，即使对于非半单Hecke代数，Mullineux映射也总是由分拆转置给出的。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经典型Hecke代数的表示型结果发表了15次演讲。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