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Remark on modular representations of non-commutative algebraic systems

关于非交换代数系统的模表示的评述

基本信息

批准号：
16540023
负责人：
ARIKI Susumu
金额：
$ 2.14万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540023/
关键词：
Hecke algebra Modular representation Kashiwara crystal path model degenerate BMW algebra modular branching rule DJM conjecture Hecke代数 B型Hecke代数有理Cherednik代数 affine Wenzl代数巡回Hecke環 affine Wenzl algebra cyclotomic Wenzl algebra

项目摘要

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

在代数群、量子群和共形场论的研究中出现了好几类代数，我们可以通过应用各种方法对这些代数进行详细的分析，因此存在一个我们可以称之为“特殊非交换代数”的研究领域。基于我们之前对Hecke代数模表示的研究，我们将本研究项目的目标设定为两个主题：一是将我们的技术发展到新的代数，二是解决Hecke代数模表示理论中的开放性问题。对于第一个目标，我们选取了定义在任意代数闭域上的简并仿射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合作的作品。这项工作影响了其他研究人员对仿射BMW代数的其他后续研究。在此期间，Rouquier证明了Cherednik代数…More提供了定义在复数域上的切环Hecke代数的准遗传覆盖。因此，我们有机会对Fock空间进行分类。这给了一个广阔的前景，推广了首席调查员的分解数定理。通过以上的研究进展，我们在项目的最后一年转向了与共形场理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，首席研究员证明了切环Hecke代数的模分支规则，这是研究计划中提到的预期成果之一。我们还解决了一个由迪珀、詹姆斯和墨菲提出的猜想，这个猜想已经存在了12年。这是和杰森合作的作品。回想一下，对称群的表示理论（以及A型Hecke代数）中的Mullineux猜想已经开放了很多年，最终在90年代由Kleshchev（和Brundan）解决，这是一个很大的成就。将Littelmann路径模型应用于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，Mullineux映射总是由分区的转置给出，即使对于非半简单的Hecke代数，如果我们在路径模型中工作。首席研究员在研究期间发表论文13篇（其中被评审论文8篇，翻译论文1篇），并就上述结果和经典型Hecke代数的表示类型成果进行了15次演讲。少