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

首席研究员将继续研究 涉及组合学和 代数 他将重点讨论两个问题。 第一个问题涉及新的 主要研究者的申请引起的争议 Kazhdan-Lusztig理论的一个组合内禀问题 它断言某些Hecke代数虚字符是 多项式与非负，偶单峰，整数 系数 这些结构的组合方法将 沿着对已知的 Kazhdan-Lusztig理论中的非负性结果， 在复杂的几何机械上。 第二个问题是 Macdonald的Shur函数展开 对称函数是具有非负整数多项式 系数 这个项目是关于组合学和 李代数 首席调查员关心的是 某些多项式是组合定义的， 与几何学和表现学中的结构有着深刻的联系 理论 这项工作在数学和物理学中都很重要。

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.