Representation Theoretical Methods in the Theory of Special Functions

本文主要研究对称函数理论及其在表示论和组合学中的应用。表示理论与对称函数理论之间的联系是由Frobenius映射提供的，它将群字符与对称函数联系起来。组合学在不可约物的多重性中扮演着重要的角色，这些不可约物通常是通过计算图、路径、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中心因素。在麦克唐纳多项式理论的十多年的研究中，研究者和M. Haiman已经在表示论、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，研究者发现了各种求和公式（PNAS V. 98（2001年4月）4313-4316），这些公式允许证明麦克唐纳多项式理论中的第一个重要的正结果。在与J. Haglund的共同工作中，研究者证明了一个有理函数的一个漂亮的组合公式（由J. Haglund推测），这个函数后来被称为$q,t$-Catalan。这位研究者与他的学生和同事合作，计划利用他最近发现的对称函数恒等式，对一些尚未解决的猜想进行直接攻击。对称函数理论是一个强大的符号操作工具。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而线性化。现在，对称函数理论的变换基矩阵可以用来“模拟”有限装置中无穷的存在，从而允许许多计算问题的线性化和求解。在对称函数的理论和应用方面的发现，也应该在数学的各个领域产生重大影响，在这些领域中对称函数方法已被证明是有效的。鉴于此，我们可以看到，在对称函数理论中进行研究，扩展和深化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.