The Combinatorics of Representations

摘要：表征的组合学本提案主要关注三个不同的项目，它们都与表征的组合学有关。第一个涉及到坦波利-利布和琼斯代数。坦波利-利布代数最初是作为物理状态之间的传递矩阵出现在统计力学中。后来，它们被发现为结点和连杆提供了重要的不变量。类似地，琼斯代数与环上的结和环有关。结理论的结果在DNA分析和蛋白质折叠等领域发挥着越来越重要的作用。提出的工作是研究矩阵的某些代数交换与坦波利-利布和琼斯代数。目标是了解相关的组合学及其在解决诸如结理论和群论等领域的各种问题中的应用。第二个课题研究偏序集合上的下算子和上算子。这样的算符出现在许多不同的环境中——例如，在物理学中，它们通常被解释为粒子上的创造和湮灭算符。由这些算子生成的代数揭示了关于集合的许多信息；它对基本的组合数据进行编码；它有助于理解集合上的随机游走。最后的项目研究了与扩展仿射根系统相关的超平面（镜像）的某些反射，它们的组合，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（矩形数字数组）。代数对象可以表现为晶体的对称性，也可以表现为物理系统的旋转。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，表征理论对粒子物理学、化学晶体研究和数学研究产生了巨大的影响。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。组合表示理论将具体的实现向前推进了一步，将这些表示与可被操纵和显式计数的组合对象联系起来。近年来，这门学科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称为坦波利-利布代数和各种其他相关的代数。坦波利-里布代数首先出现在统计力学中，用来描述物理状态之间的能量转移。在20世纪80年代，沃恩·琼斯证明了它们与结的研究有关。通过研究它们的各种特性，该项目寻求开发区分结和连接的新方法。这在DNA分析和蛋白质折叠等领域具有潜在的应用前景。这些项目的一些组成部分可以由本科生和研究生初学者承担，因为组合学的具体性质为学生提供了一个很好的工具来介绍研究。首席研究员乔治亚·本卡特（Georgia Benkart）认为，让学生接触数学研究，让他们相信自己能够理解并积极参与研究，这一点很重要。学生将以基本的方式参与项目，并将开展与这里提出的项目相关的自己的研究项目。

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.