The Combinatorics of Representations

表示组合学

• 批准号: 0245082
• 负责人: Georgia Benkart
• 金额: $ 14.08万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245082&HistoricalAwards=false
• 关键词: Combinatorics; Representations

Principal Investigator: Georgia Benkart Proposal Number: 0245082Institution: University of Wisconsin-MadisonAbstract: The combinatorics of representations This proposal focuses on three different projects -- all related to the combinatorics of representations. The first involves Temperley-Lieb and Jones algebras. Temperley-Lieb algebras appeared initially in statistical mechanics as transfer matrices between physical states. Later they were discovered to provide important invariants of knots and links. Similarly, the Jones algebras are related to knots and links on an annulus. Results in knot theory are playing an ever more significant role in such topics as DNA analysis and protein folding. The proposed work is to study certain algebras of matrices that commute with the Temperley-Lieb and Jones algebras. The goal is to understand the associated combinatorics and its applications in addressing a variety of problems in areas such as knot theory and group theory. The second project studies down and up operators on sets with a partial order. Such operators have appeared in many different contexts -- for example, in physics where they are often interpreted as creation and annihilation operators on particles. The algebra generated by these operators reveals much information about the set; it encodes essential combinatorial data; and it contributes to the understanding of such things as random walks on the set. The final project investigates certain reflections in hyperplanes (mirrors) related to extended affine root systems, their combinatorics, and their actions on various spaces. All the projects involve the representation theory of algebras and groups.The goal of representation theory is to ``represent'' an abstract algebraic object as explicit matrices (rectangular arrays of numbers) that describe its action on a space. The algebraic object might be acting as the symmetries of a crystal or as rotations of a physical system. Representation theory has had an enormous impact on particle physics, on the study of crystals in chemistry, and on mathematical research ever since the pioneering work of mathematician Issai Schur and physicist Hermann Weyl in the 1920's. Its continuing vitality is evidenced by much current activity and many open problems. Combinatorial representation theory takes the concrete realization one step further by associating to such representations, combinatorial objects that can be manipulated and counted explicitly. The subject has had an explosion of activity in recent years with numerous important applications in diverse areas of mathematics and physics. This proposal seeks to understand the combinatorics of certain algebras called Temperley-Lieb algebras and various other related algebras. Temperley-Lieb algebras first arose in statistical mechanics where they were used to describe the transfer of energy between physical states. In the 1980's, Vaughan Jones showed that they are related to the study of knots. By studying various properties of them, the project seeks to develop new ways of distinguishing knots and links. This has potential applications to such subjects as DNA analysis and protein folding. Some components of these projects can be undertaken by undergraduate and beginning graduate students, as combinatorics provides an excellent vehicle to introduce students to research because of its concrete nature. The principal investigator, Georgia Benkart, feels it is important to expose students to mathematical research and to convince them that they can understand and take an active role in research. Students will participate in the project in essential ways and will conduct their own research projects related to the ones proposed here.

主要研究者：格鲁吉亚Benkart提案编号：0245082机构：威斯康星大学麦迪逊分校摘要：表示的组合学 这个建议集中在三个不同的项目-所有相关的组合表示。 第一个涉及Temperley-Lieb和Jones代数。 坦珀利-李伯代数最初出现在统计力学中，作为物理状态之间的转移矩阵。 后来发现它们提供了纽结和链环的重要不变量。 类似地，琼斯代数与环上的纽结和链环有关。 结理论的结果在DNA分析和蛋白质折叠等课题中发挥着越来越重要的作用。 拟开展的工作是研究与Temperley-Lieb和Jones代数可交换的矩阵的某些代数。目标是了解相关的组合学及其在解决纽结理论和群论等领域的各种问题中的应用。 第二个项目研究了偏序集上的上下运算符。 这样的算子出现在许多不同的背景下-例如，在物理学中，它们通常被解释为粒子的创造和湮灭算子。 由这些算子生成的代数揭示了集合的许多信息;它编码了基本的组合数据;它有助于理解集合上的随机游动等问题。 最后一个项目研究了与扩展仿射根系统相关的超平面（镜子）中的某些反射，它们的组合学以及它们在各种空间上的作用。 所有的项目都涉及代数和群的表示理论。表示理论的目标是将抽象的代数对象“表示”为描述其在空间上的作用的显式矩阵（数字的矩形阵列）。 代数对象可以作为晶体的对称性或作为物理系统的旋转。 表示论对粒子物理学、化学中晶体的研究和数学研究产生了巨大的影响，这是自20世纪20年代数学家Issai Schur和物理学家Hermann Weyl的开创性工作以来的。目前的许多活动和许多悬而未决的问题证明了它的持续活力。 组合表示理论通过将可以被显式地操作和计数的组合对象与这样的表示相关联，使具体实现更进一步。 近年来，这一主题在数学和物理的各个领域都有许多重要的应用。这个建议试图理解某些代数的组合学，称为Temperley-Lieb代数和各种其他相关的代数。 坦珀利-李伯代数首先出现在统计力学中，它们被用来描述物理状态之间的能量转移。在20世纪80年代，沃恩·琼斯（Vaughan Jones）表明它们与结的研究有关。 通过研究它们的各种属性，该项目寻求开发区分结和链接的新方法。 这对DNA分析和蛋白质折叠等学科具有潜在的应用价值。 这些项目的一些组成部分可以由本科生和开始研究生进行，因为组合学提供了一个很好的工具，介绍学生的研究，因为它的具体性质。 首席研究员，格鲁吉亚本卡特，认为这是重要的是让学生接触到数学研究，并说服他们，他们可以理解，并采取积极的作用，在研究。 学生将以重要的方式参与该项目，并将进行与本文提出的项目相关的研究项目。