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

代数组，量子群和保形场理论的研究中出现了几种代数，因为我们可以通过应用各种方法对这些代数进行详细分析，存在一个研究领域，我们可能称之为“特殊非共同代数”。基于我们先前对Hecke代数模块化表示的研究，我们以两个主题设定了该研究项目中的目标：第一个是向新代数的技术开发我们的技术，第二个是解决Hecke代数模块化代表理论中的开放问题。对于第一个目标，我们选择了根据任意代数封闭的封闭场定义的退化仿射BMW代数，并成功地构建了所有不可还原的有限维度表示。这是与Mathas和Rui的联合工作。这项工作影响了其他研究人员对仿射宝马代数的其他成功研究。在此期间，鲁奎尔（Rouquier）表明，Cherednik代数…更多提供了在复数领域定义的循环组合hecke代数的准植物覆盖物。那就是我们有机会对fock空间进行分类。这给出了广泛的视角，从而概括了主调查员的分解编号定理。通过上述研究的发展，我们已经转向了研究的研究理论在项目的最后一年中更加紧密相关，并且我们为下一个研究项目获得了新的见解。对于第二个目标，首席调查员已证明了Cyclotomic Hecke代数的模块化分支规则，该代数在研究建议中被提及是预期的成就之一。我们还解决了Dipper，James和Murphy的猜想，该猜想已经开放了12年。这是与雅各共同的合作。回想一下，对称群体的表示理论（以及A型Hecke代数）在90年代的Kleshchev（和Brundan）最终解决，这是一个很大的成就。通过将Littelmann的路径模型应用于Hecke代数的表示理论，我们获得了著名的Mullineux图的全新描述。也就是说，如果我们在路径模型中工作，即使对于非半个半普通代数的分区，也会给予mullineux图。首席调查员已经发表了13篇论文（其中8篇论文是参考论文的翻译），并就上述结果进行了15次谈判，并在研究期间就经典类型的Hecke代数的代表类型进行了讨论。较少的