Combinatorics of the Affine Hecke Algebra and Module Categories

仿射赫克代数和模范畴的组合

• 批准号: 0535944
• 负责人: Victor Ostrik
• 金额: $ 3.57万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-05-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0535944&HistoricalAwards=false
• 关键词: Combinatorics; Affine; Hecke; Algebra; Module

The proposal consists of 4 parts. In the first part the investigator and his colleagues study the asymptotic affine Hecke algebra introduced by G. Lusztig. The asymptotic Hecke algebra is a suitable limit of the usual Hecke algebra as parameter tends to zero. Its representation theory is closely related with representation theory of the Hecke algebra itself. One of the aims here is a proof of Lusztig's Conjecture describing the asymptotic Hecke algebra in the elementary K-theoretic terms. In the second part the investigator studies module categories over monoidal categories. This subject is closely related with modern physics where module categories appear in the context of the Boundary Conformal Field Theory. In the third part the investigator and collaborators study distinguished involutions in the affine Weyl group. In particular they make extensive explicit calculations of canonical distinguished involutions in number of cases. In the fourth part the investigator and his colleagues study the Double Affine Hecke Algebra. The aim here is to describe Intersection Cohomology of certain infinite dimensional algebraic varieties in terms of Kazhdan-Lusztig type combinatorics of this algebra.In this proposal the investigator studies various questions of Representation Theory. Representation Theory is a part of mathematics that studies all possible ways in which symmetry can be used for solving concrete physical or technical problems. Many physical and technical systems do not change under some transformations (e.g. the gravitational field of the Sun depends only on the distance from the Sun and so it does not change under rotating of the space around the Sun). Such transformations are called symmetries of the system. In many cases symmetries can be used to simplify the study of such systems. So it is not surprising that Representation Theory has many applications in physics (where continuous symmetry is one of the most fundamental concepts), chemistry (especially in quantum chemistry where it is used in computations of chemical forces inside molecules), computer science (for example, Fourier analysis, which can be considered as a simplest case of Representation Theory, is one of the most widely used of all calculation techniques), and inside of mathematics itself, in number theory (where Representation Theory is an essential part of the Langlands program). One of the central objects of study in Representation Theory is the affine Hecke algebra, because answers to many seemingly unrelated questions are encoded in the structure of this algebra. This proposal is mainly devoted to the study of the affine Hecke algebra.

该提案包括四个部分。第一部分研究了G.卢斯蒂格 渐近Hecke代数是通常Hecke代数在参数趋于零时的一个合适的极限。 它的表示理论与Hecke代数本身的表示理论密切相关。 本文的目的之一是证明Lusztig猜想描述的渐近Hecke代数的基本K理论条款。 在第二部分中，研究者在monoidal范畴上研究模范畴。 这门学科与现代物理学密切相关，其中模范畴出现在边界共形场论的背景下。 在第三部分中，研究者和合作者研究了仿射Weyl群中的对合。 特别是，他们对许多情况下的典型可区分对合进行了广泛的显式计算。 第四部分研究了双仿射Hecke代数。 这里的目的是描述相交上同调的某些无限维代数簇的Kazhdan-Lusztig型组合代数的代数。在这个建议中，研究者研究的各种问题的表示论。表示论是数学的一部分，研究对称性可以用于解决具体物理或技术问题的所有可能方式。 许多物理和技术系统在某些变换下不会改变（例如，太阳的引力场只取决于到太阳的距离，因此它不会在围绕太阳旋转的空间中改变）。 这种变换称为系统的对称性。 在许多情况下，对称性可以用来简化此类系统的研究。 所以表示论在物理学中有许多应用也就不足为奇了（其中连续对称是最基本的概念之一），化学（特别是在量子化学中，它用于计算分子内部的化学力），计算机科学（例如，傅立叶分析，它可以被认为是表示论的最简单的情况，是所有计算技术中使用最广泛的一种），在数学本身内部，在数论中（表示论是朗兰兹纲领的重要组成部分）。在表示论的研究的中心对象之一是仿射赫克代数，因为许多看似无关的问题的答案编码在这个代数的结构。 本文主要研究仿射Hecke代数。