Combinatorics of Young Tableaux and their Generalizations

调查者将研究与有限和无限维李群的表示有关的组合问题。其目的是获得表示的显式组合描述，并反过来使用表示理论来回答组合问题。首先，他将研究斯珀纳理论，该理论研究以“不干涉”的方式组织集合和数字的方法。研究人员将使用一种新的与量子群表示的晶基的联系来解决涉及偏序集的对称链分解的组合问题。接下来，他的目标是应用组合学，特别是利特尔曼和京都路径模型，来控制和澄清环群表示的代数结构。最后，研究人员将探索组合和几何表示理论对构型变种、多旗变种和箭图表示的推广。表示理论是对称对象理论。由于对称性通常是在复杂情况下获得精确解的关键，表示理论是数学、物理和化学中最强大的工具之一。这位研究人员的工作试图用明确的组合术语来描述各种对称性。简而言之，目标是理解n维(和无限维)空间中的对象，这些对象像圆或球体一样，具有一大族连续的对称性，并将这种对称性编码到离散组合学中。

DMS-0405948Peter M. MagyarThe investigator will examine combinatorial problems related to representations of Lie groups, both finite and infinite-dimensional. The aim is to gain an explicit combinatorial description of the representations, and conversely to use representation theory to answer combinatorial questions. First he will work in Sperner Theory, which examines methods to organize sets and numbers in ``non-interfering'' ways. The investigator will use a new connection with crystal bases of quantum group representations to address combinatorial problems involving Symmetric Chain Decomposition of posets. Next, he aims to apply combinatorics, specifically the Littelmann and Kyoto path models, to control and clarify the algebraic structure of loop group representations. Finally, the investigator will explore generalizations of combinatorial and geometric representation theory to configuration varieties, multiple flag varieties, and quiver representations.Representation theory is the theory of symmetric objects. Since symmetry is usually the key to obtaining exact solutions in complicated situations, representation theory is one of the most powerful tools in mathematics, physics, and chemistry. The investigator's work seeks to describe various kinds of symmetry in explicit combinatorial terms. Put simply, the aim is to understand the objects in n-dimensional (and infinite-dimensional) space, which, like the circle or sphere, have a large, continuous family of symmetries, and to encode this symmetry in discrete combinatorics.