Mathematical Sciences: Constant Term Identities and Schur Functions

这项研究将集中在三个领域： q-MacDonald-Morris常数项证明的完善 本文给出了一些结果的组合证明， B根系和q-Selberg多项式。 基本主题是 A根成功的技术的推广 其他根系统。 卡德尔已经扩展了Aomoto的 证明Selberg积分的q-的情况下，并证明了q- BC根系的MacDonald-Morris猜想 这应该 延伸到治疗剩余的根系。 一些成果 对于与A根系统相关的Schur函数， 组合证明。 卡德尔将攻击一些相关的问题 与B根系相关。 最后是Q-塞尔伯格 多项式扩展了Schur函数，小q-Jacobi 多项式和带状多项式，希望 这些多项式的理论，包括组合 表示，可以基于约束正交性 关系。 本研究的重点是组合理论 恒等式是一个重要的学科， 有深刻问题的想法和代数技巧， 分析. 这个问题最近变得更加重要 这一发现与物理学有着重要的联系。 麦克唐纳 它们的许多变体一直是中心焦点 多年来，这一地区。 卡德尔正在 在这些深刻的问题上取得进展，并将继续取得进展， 在这项研究中做出了令人兴奋的贡献。

This research will concentrate on three areas: the completion of the proof of the q-MacDonald-Morris constant term conjectures, combinatorial proofs of some results related to the B root system and the q-Selberg polynomials. The basic theme is the extension of techniques which are successful for the A root system to the other root systems. Kadell has extended Aomoto's proof of Selberg's integral to the q-case and proved the q- MacDonald-Morris conjecture for BC root system. This should extend to treat the remaining root systems. A number of results for Schur functions related to the A root systems have been proven combinatorially. Kadell will attack some related problems associated with the B root system. Finally the q-Selberg polynomials extend the Schur functions, the little q-Jacobi polynomials and the zonal polynomials and it is hoped that a theory of these polynomials, including a combinatorial representation, can be based on a conjectured orthogonality relation. This research focuses on the theory of combinatorial identities, an important subject that combines combinatorial ideas with deep problems and techniques from algebra and analysis. The subject has recently become even more important with the discovery with important ties to physics. The MacDonald conjectures and their many variants have been the central focal point of this area for years. Kadell is making significant progress on these deep problems and will continue to make exciting contributions during this research.