Representation Theoretical Methods in the Theory of Special Functions

这个项目分为两个独立的主要研究领域。m-拟不变量的一般理论与麦克唐纳多项式理论。准不变量最初出现在粒子动力学的研究中，但已经成为一个真正迷人的代数组合学科。对于每一个Coxeter群G，在普通多项式环和G不变量环之间都有一个相关联的1个参数子空间族。$G$的m-拟不变量是简单的多项式，在对应于G的一个反射超平面的差分算子的作用下，它们产生一个因子，该因子可被反射超平面的2m+1次幂的方程整除。从代数的角度来看，这一理论在Feigin-Veselov和Felder-Veselov的基础论文中得到了广泛的发展，并在2002年由Etingov-Ginsburg证明的某些猜想中达到顶峰。与N. Wallach一起，PI发现了与$m$-拟不变量相关的几个结构，例如相关的Baker-Achieser函数，编码了相应cox - ter群的一些真正显著的组合特性。本课题就是从组合的角度来探讨这一理论。在二面体群的最简单情况的研究中获得的早期成功强烈地表明，这样的研究应该有助于在数学的各个领域，从组合学，表示理论和特殊函数理论的重大发现。自1988年被发现以来，麦克唐纳对称函数基逐渐成为表征理论和对称函数理论之间联系的中心元素。在对麦克唐纳多项式理论进行了十多年的研究之后，PI、M. Haiman及其合作者在表示论、代数几何、组合学和对称函数理论中提出了各种各样的猜想。在证明这些猜想的努力中，在每个领域都产生了基本的事实和方法。最近，J. Haglund发现并与Haiman和Loehr合作证明了一个值得注意的纯组合公式，用于适当修正的麦克唐纳多项式族。这一发现提出了一系列的研究问题，涉及到一些直到最近才出现的猜想。Pi计划与他的博士生合作，在这一领域投入大量的研究工作。提议的工作，涉及广泛的计算机探索和跨越几个数学边界的信息传递，从表示理论到理论特殊函数，然后到组合学。这些转移提供了一种宝贵的发现工具，因为在其中一个领域中可能非常明显的结果和机制通常会在另一个领域转化为非常重要和意想不到的事实。应该提到的是，像MAPLE和MATHEMATICA这样的符号处理软件与当前一代的快速处理器相结合，确实将数学的许多分支转变为实验科学。此外，计算机数据不仅可以成功地用于结果的发现，而且还可以用于证明的构建。这两个基本项目是计划研究的重点，也非常适合培养年轻研究人员，并为他们提供机会，让他们发现和体验在21世纪进行研究的方式。因此，这个项目产生的所有研究问题都会被带到课堂上，与学生和合作者分享。

This project is in two separate major research areas. The generalTheory of m-Quasi-invariants and the Theory of Macdonald Polynomials.m-Quasi-invariants originally arose in the study of particle dynamicsbut have emerged as a truly fascinating Algebraic Combinatorialsubject. For each Coxeter Group G there is an associated one parameterfamily of subspaces which interpolates between the ordinary polynomial ring and the ring of G-invariants. The m-Quasi-Invariants of $G$ are simply polynomials which, upon the action of the difference operator corresponding to a reflecting hyperplane of G, they produce a factor that is divisible by the equation of the reflecting hyperplane raised to the power 2m+1. This theory has been extensively developped, from the algebraic point of view, in fundamental papers by Feigin-Veselov, and Felder-Veselov, and culminated in certain conjectures which were proved by Etingov-Ginsburg in 2002. Jointly with N. Wallach the PI discoveredthat several constructs associated to $m$-quasi-invariants,such as the associated Baker-Achieser function, encode some trulyremarkable combinatorial properties of the corresponding Coxeter group.The present project is to explore the theory from the combinatorialpoint of view. The early successes obtained in the study of the simplest cases of the dihedral groups strongly suggest that such a study should be conducive to significant discoveries in various areas of Mathematics ranging from Combinatorics, Representation Theory and the Theory of special functions.Since its 1988 discovery, the Macdonald symmetric function basis hasprogressively emerged as a central element in the connection betweenRepresentation Theory and the Theory of Symmetric Functions.After more than a decade of research in the Theory of Macdonald polynomials, the PI, M. Haiman and collaborators have been led to a variety of conjectures in Representation Theory, Algebraic Geometry, Combinatorics and Symmetric Function Theory. Efforts in proving these conjectures have yielded fundamental facts and methods in each of these areas. Recently J. Haglund discovered and in collaboration with Haiman and Loehr proved a remarkable purely combinatorial formula for a suitably modified family of Macdonald polynomials. This discovery opens up a variety of research problems involving some of the conjectureswhich up to quite recently appeared inaccesible. The Pi plans to dedidcatea substantial portion of the projected research effort in this area incollaboration with his PhD students. The proposed work, involves extensive computer explorations andtransfer of information across several mathematical boundaries, fromRepresentation Theory to the Theory Special Functions and then to Combinatorics. These transfers provide an invaluable vehicle of discovery, since results and mechanisms which may be quite obvious in one of these areas often translate in highly non trivial and unexpected facts in one of the other areas. It should be mentioned that symbolic manipulation software such as MAPLE and MATHEMATICA combined with the present generation of fast processors have literally transformed many branches of mathematics into experimental sciences. Moreover computer data can be successfully used not only in the discovery of results but also in the construction of proof. The two basic projects that are the focus of the planned research are also highly suitable for the training of young researchers and the opportunity that it provides them to discover and experience the manner in which research can be carried out in the 21st century. Thus all the research problems created by this projectare invariably brought in the the class room to be shared with studentsand collaborators.