Coxeter群上Kazhdan-Lusztig多项式的组合研究

Kazhdan-Lusztig理论一经提出, 其上的关键结构Kazhdan-Lusztig多项式就得到了广泛的关注. 由于Coxeter群所具有的组合背景, 从组合角度研究Kazhdan-Lusztig多项式成为了代数组合学的一个重要课题. 著名组合学家F. Brenti曾猜测多数Kazhdan-Lusztig多项式具有封闭的表达式或良好的生成函数. 本项目计划从以下几个方面对Kazhdan-Lusztig多项式进行组合研究: 一, 从Coxeter群上经典组合结构入手, 研究计算Kazhdan-Lusztig多项式的特殊表达式. 二, 从Kazhdan-Lusztig多项式的组合解释入手, 给出其上新的组合性质或对已有代数性质进行组合证明. 三, 对Coxeter群上的偏序结构进行研究, 希望在Kazhdan-Lusztig多项式组合不变性猜想上有所进展.

Since the Kazhdan-Lusztig theory was established, the Kazhdan-Lusztig polynomials have gained much attention. Because the theory of Coxeter groups stands a strong combinatorial background, combinatorics of Kazhdan-Lusztig polynomials have become important topics in algebraic combinatorics. As a well-known combinatorialist, F. Brenti had conjectured that many kinds of Kazhdan-Lusztig polynomials should be well expressed or generated. This project mainly focuses on the following problems: 1. Express or generate some kinds of Kazhdan-Lusztig polynomials, by engaging the combinatorial structures of Coxeter groups; 2. Discover or rediscover the properties of Kazhdan-Lusztig polynomials from combinatorial aspects, by focusing on the combinatorial interpretations of Kazhdan-Lusztig polynomials; 3. Investigate the partial order structure of Coxeter groups and make progress on the combinatorial invariance conjecture of Kazhdan-Lusztig polynomials.