Mathematical Sciences: Constant Term Identities and Schur Functions

这项研究将集中在三个方面：q-MacDonald-Morris常数项猜想的证明的完成，与B根系和q-Selberg多项式有关的一些结果的组合证明。其基本主题是将A根系成功的技术推广到其他根系。Kadell将Aomoto关于Selberg积分的证明推广到Q-情形，并证明了BC根系统的Q-MacDonald-Morris猜想。这应该延伸到治疗剩余的根系统。与A根系统有关的Schur函数的一些结果已被组合证明。卡德尔将解决一些与B根系统相关的问题。最后，q-Selberg多项式推广了Schur函数、小q-Jacobi多项式和带状多项式，并希望这些多项式的理论，包括组合表示，可以基于猜想的正交性关系。这项研究的重点是组合恒等式理论，这是一门将组合思想与代数和分析中的深层问题和技巧相结合的重要学科。最近，随着与物理学有重要联系的发现，这个问题变得更加重要。多年来，麦克唐纳猜想及其许多变体一直是这一领域的中心焦点。卡德尔在这些深层次问题上取得了重大进展，并将在这项研究中继续做出令人振奋的贡献。

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.