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

首席研究员将继续他对涉及组合学和代数之间相互作用的问题的研究。他将专注于两个问题。第一个涉及首席研究者将Kazhdan-Lusztig理论应用于一个关于组合内蕴问题的新猜想，该问题断言某些Hecke代数虚拟特征标是具有非负甚至单峰整数系数的多项式。将寻求对这些猜想的组合方法，以及对已知的非负性结果的组合解释，从而形成Kazhdan-Lusztig理论，该理论现在依赖于困难的几何机械。第二个问题是关于Macdonald对称函数的Shur函数展开式是非负整系数多项式的猜想。这个项目是关于组合学和李代数之间的接口的。原理研究者关注的是某些组合定义的多项式，这些多项式在几何和表示理论上与结构有很深的联系。这项工作在数学和物理中都很重要。

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.