Representation Theory and Combinatorics

该建议的研究问题是在组合表示理论领域，并有，作为主要目标，组合结构的不可约表示及其字符。 代数结构下的研究是所有的群体和代数来自，或推广，李理论的对象。 该提案的主要资金要求是支持主要研究员的三名研究生。研究生研究项目集中在（1）环形李代数，环形量子群和环形Yangians，（2）复反射群和（3）Yokonuma-Hecke代数的组合表示理论。 本论文的主要研究方向是仿射Hecke代数和Iwahori-Hecke代数的组合表示理论。 这项研究强烈使用代数几何和组合工具。大多数有趣的对称性是足够复杂的，很难直接研究它们，表示是提取这些对称性信息的一种方式。于是就诞生了“表象理论”。 然而，通常表示本身是相当复杂和复杂的，组合表示理论的目标是找到基本模型，使我们能够更容易地确定这些表示的属性：大小，组件数量，拆分和组合规则以及字符。 最有用的模型类型具有儿童游戏的味道，如LEGOS或ERECTOR集，但这些模型使人们能够获得有关相应表示的精细结构的非常明确的信息。与分子生物学类似，一般表示是一个大而复杂的结构，如分子，由较小的“不可约成分”组成，类似于原子。 在这个建议的研究是针对确定的属性的不可约的组成部分，并确定哪些不可约的组成部分出现在自然发生的“标准”的表示。

The proposed research problems of this proposal are in the field of combinatorial representation theory and have, as primary goals, combinatorial constructions of irreducible representations and their characters. The algebraic structures under study are all groups and algebras which come from, or are generalizations of, Lie theoretic objects. The primary funding request of this proposal is for support for three graduate students of the principal investigator. The graduate student research projects center on the combinatorial representation theory of (1) toroidal Lie algebras, toroidal quantum groups and toroidal Yangians, (2) complex reflection groups and (3) Yokonuma-Hecke algebras. The research projects of the proposer are focused on the the combinatorial representation theory of affine Hecke algebras and Iwahori-Hecke algebras. This research strongly uses both algebraic geometric and combinatorial tools.Most interesting symmetries are complex enough that it is difficult to study them directly and a representation is a way of extracting information about these symmetries. Thus is born ``representation theory''. However, very often the representations themselves are quite intricate and complicated and the goal of Combinatorial Representation Theory is to find elementary models which allow us to more easily determine properties of these representations: size, number of components, splitting and combination rules, and character. The type of models that are the most useful have the flavor of games for children, like LEGOS or ERECTOR sets, and yet these models enable one to obtain very explicit information about the fine structure of the corresponding representations. In analogy with molecular biology, a general representation is a large and complex structure like a molecule and is composed of smaller ``irreducible components'', analogous to atoms. The research in this proposal is directed towards determining the properties of the irreducible components and on determining which irreducible components appear in naturally occuring ``standard'' representations.