Mathematical Sciences: Combinatorial Methods in Algebra: Coxeter Groups, Hecke Algebras, Young Tableaux, and Symmetric Functions

数学科学：代数组合方法：Coxeter 群、Hecke 代数、Young Tableaux 和对称函数

• 批准号: 9119355
• 负责人: Mark Haiman
• 金额: $ 4.65万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-01-01 至 1994-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9119355&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Combinatorial; Methods; Algebra

The principle investigator will continue his research on problems involving the interaction between combinatorics and algebra. He will focus on two problems. The first concerns new conjectures arising from the principle investigator's application of Kazhdan-Lusztig theory to a problem on combinatorial immanants which assert that certain Hecke algebra virtual characters are polynomials with with non-negative, even unimodal, integer coefficients. A combinatorial approach to these conjectures will be sought, along with combinatorial interpretations of known non-negativity results in Kazhdan-Lusztig theory which now rest on difficult geometric machinery. The second problem concerns the conjecture that the Shur function expansion of Macdonald's symmetric functions are polynomials with non-negative integer coefficients. This project is on the interface between combinatorics and Lie algebra. The principle investigator is concerned with certain polynomials which are combinatorially defined and have deep connections with structures in geometry and representation theory. This work is important both in mathematics and physics.

首席研究员将继续研究涉及组合学和代数相互作用的问题。他将关注两个问题。第一部分涉及主要研究者将Kazhdan-Lusztig理论应用于组合内因式问题所产生的新猜想，该问题断言某些Hecke代数虚字符是具有非负偶数单峰整数系数的多项式。将寻求对这些猜想的组合方法，以及对Kazhdan-Lusztig理论中已知的非负性结果的组合解释，这些结果现在依赖于困难的几何机制。第二个问题涉及麦克唐纳对称函数的舒尔函数展开式是非负整数系数多项式的猜想。这个项目是关于组合学和李代数之间的接口。主要研究组合定义的多项式，这些多项式与几何和表示理论中的结构有很深的联系。这项工作在数学和物理学上都很重要。