Representation Theoretical Methods in the Theory of Special Functions

本文是关于对称函数理论及其在表示论和组合学中的应用的研究。通过将群特征标与对称函数联系起来的Frobenius映射，提供了表示论与对称函数论之间的联系。组合数学的作用在于，不可约的重数通常是通过计算表、路、树和越来越多的新出现的离散结构来获得的。自发现以来的十多年里，麦克唐纳基础逐渐成为这些联系的核心要素。经过十多年对麦克唐纳多项式理论的研究，研究者和M.Haiman在表象理论、代数几何、组合学和对称函数理论中得出了各种猜想。证明这些猜想的努力在每个领域都产生了基本的事实和方法。最近，研究人员发现了各种求和公式(PNAS V.98(4月2001)4313-4316)，这些公式允许证明麦克唐纳多项式理论中的第一个显著的正性结果。在与J.Haglund的合作中，研究人员证明了一个被称为$q，t$-加泰罗尼亚的有理函数的美丽组合公式(J.Haglund猜想)。这位研究人员与学生和同事合作，计划利用他最近发现的对称函数恒等式来直接攻击一些尚未解决的猜想。对称函数理论是一个强大的符号处理工具。这样做的原因是，非线性问题通常可以通过引入无限数量的变量来线性化。现在它发展到对称函数理论的基矩阵的变化可以用来“模拟”有限设备中无穷大的存在，从而允许许多计算问题的线性化和解决。在对称函数的理论和应用方面的发现，也应该在对称函数方法被证明是有效的数学的各个领域中产生重大影响。考虑到这一点，我们可以看出，继续研究对称函数理论，扩展和深化该理论的计算能力是多么重要。这是本项目的首要目标。

This research is on the Theory of Symmetric Functions and its applications to Representation Theory and Combinatorics. The connection between Representation Theory and the Theory of Symmetric Functions is provided by the Frobenius map which relates group characters to symmetric functions. Combinatorics plays a role in that multiplicities of irreducibles are often obtained by counting tableaux, paths, trees and a growing variety of newly emerging discrete structures. In the more than ten years since its discovery, the Macdonald basis has progressively emerged as a central element in these connections. For more than a decade of research in the Theory of Macdonald Polynomials, the investigator and M. Haiman 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. More recently the investigator discovered a variety of summation formulas (PNAS V. 98 (April 2001) 4313-4316) which permitted the proof of the first significant positivity result in the Theory of Macdonald Polynomials. In joint work with J. Haglund the investigator proved a beautiful combinatorial formula (conjectured by J. Haglund) for a rational function which had come to be known as the $q,t$-Catalan. The investigator in collaboration with students and associates plans to use his recently discovered symmetric function identities for a direct attack of some of the conjectures that are still unresolved.The Theory of Symmetric Functions is a powerful symbolic manipulation tool. The reason for this is that non linear problems may often be linearized by the introduction of an infinite number of variables. Now it develops that the change of bases matrices of Symmetric Function Theory may be used to "mimic" the presence of infinities within a finite device, thereby permitting the linearization and solution of many a computational problem. Discoveries in the theory and applications of symmetric functions, should also turn out to be of significant impact in the various areas of mathematics in which symmetric function methods have been shown to be effective. This given, we can see how important it is to pursue investigations in the Theory of Symmetric functions that extend and deepen the computational power of the theory. This is the foremost goal of the present project.