Irreducible Representations of the Affine and Double Affine Hecke Algebras of Type A

A型仿射和双仿射Hecke代数的不可约表示本项目的目的是对仿射和双仿射Hecke代数的不可约模进行参数化、构造和描述。研究人员打算从这些代数的非半单表示中提取组合结构，其中一个重要的例子是A型仿射Hecke代数不可约表示上的晶图结构。Hecke代数的表示理论在数学的许多领域都有应用。Hecke代数自然而然地出现在半单p-add群的表示理论中，也是研究有限域上约化群的模表示理论的一个工具。它们与量子群、统计力学和纽结理论有着密切的联系。双仿射Hecke代数是由Cherednik定义的，并被他用来证明Macdonald的某些猜想。它们与对称空间的调和分析以及经典的超几何函数和q-超几何函数理论有联系。这是表示论中的一个项目，它是对对称性的研究。表示论给我们提供了解决任何对称系统问题的工具，因此在化学、物理、计算机科学，甚至在数学的其他领域都有广泛的应用。研究人员将研究最基本的物体，这些物体的对称性被编码在一种称为黑克代数的结构中。她的主要方法之一是将这些基本物体以一种明显的全局控制结构的方式粘合在一起-就像将元素组织到元素周期表中给我们提供关于卤素(或其他基团)的共同化学性质的信息一样。

Irreducible representations of the affine and double affine Hecke algebras of type AThe aim of this project is to parameterize, construct, and describe the irreducible modules of the affine and double affine Hecke algebras. The investigator intends to extract combinatorial structure from the non-semisimple representations of these algebras, a key example of which is the crystal graph structure on the irreducible representations of the affine Hecke algebra of type A. The representation theory of Hecke algebras has applications in many areas of mathematics. Hecke algebras appear naturally in the representation theory of semisimple p-adic groups and are also a tool in the study of the modular representation theory of reductive groups over finite fields. They have intimate connections to quantum groups, statistical mechanics, and knot theory. Double affine Hecke algebras were defined by Cherednik and used by him to prove certain conjectures of Macdonald. They have connections to harmonic analysis of symmetric spaces and the classical theories of hypergeometric functions and q-hypergeometric functions.This is a project in representation theory, which is the study of symmetry. Representation theory gives us the tools to solve problems about any system that exhibits symmetry, and so has wide applications in chemistry, physics, computer science, and even within other areas of mathematics. The investigator will study the most basic objects whose symmetries are encoded in a structure called the Hecke algebra. One of her primary methods is to "glue" these basic objects together in such a way that their global controlling structure is apparent---much in the same way that organizing the elements into the periodic table gives us information about the shared chemical properties of halogens (or other groups).