Representation Theoretical Methods in the Theory of Special Functions

提出的研究跨越表示理论，对称函数理论和组合学。Frobenius映射是表示理论和组合学之间的桥梁，它将维数和多重性问题编码为与不同对称函数基相关的多项式的系数；后者已被证明可以计算各种组合结构，如tableaux，晶格路径和树状结构。1987年，Macdonald引入了一种新的对称函数基，通过它与表示论、代数组合学、代数几何、粒子物理和统计学的深刻联系，深刻地丰富了对称函数理论。1988年，PI在一个证明q，t-Kotska多项式的正性的程序中提出了麦克唐纳的基础与表征理论的联系（这是麦克唐纳论文中真正具有开创性的猜想）。PI的程序是为了证明麦克唐纳多项式的某些修正形式是在Frobenius映射下，某些梯度模的特征的图像；然后，q，t-Kotska将在这些模块的各种双齐次子空间中产生不可约表示的多重性。1990年，PI和Mark Haiman将这些重阶模构造为对角谐波的子模，并证明了q，t-Kotska猜想的成立，如果对n的每一次划分，对应的模可以被证明具有n！这后来被称为n！-猜想，2001年由Mark Haiman在Hilbert格式的代数几何上进行了十年的深入研究后证明。提出的研究是研究对角谐波表示理论、麦克唐纳多项式理论和计算机科学中所谓的停车函数之间的联系。该程序是对2002年Shuffle猜想的三管齐下的攻击，该猜想将对角谐波特征的Frobenius图像表达为停车函数的加权和。PI计划在项目的表示理论部分工作，并指导他的博士生进行对称函数理论和主题的组合部分。在麦克唐纳论文发表后的几十年里，PI、他的合作者和他的学生在麦克唐纳多项式理论中获得了大量的结果，这些结果已经证明了Shuffle猜想的各种特殊情况。无论当前项目的结果如何，正如研究困难的数学问题时常见的那样，所提出的工作将导致在意义上甚至可能超过原问题的解决方案的发现。在这项拨款下进行的研究涉及将信息从纯代数结构转移到明确的对称函数，并最终转移到组合对象，如表、路径和树。所提出的代数问题只能通过对称函数理论的进展来解决。对称函数，反过来，是一个广泛适用的计算设备。因此，拟议研究的进展应扩大物理学家和工程师获得科学应用所需的硬数据的各种工具。所有提议的研究都是在计算机已经转变为实验科学的数学领域。在这样的环境中，即使是初学的学生也能体验到非平凡发现的乐趣。因此，这项工作是一个理想的环境，可以向我们的新一代研究人员传达对计算机指导研究提供的广泛可能性的更深层次的理解。

The proposed research extends across Representation Theory, the Theory of Symmetric Functions and Combinatorics. The bridge between Representation Theory and Combinatorics is provided by the Frobenius map, which encodes dimension and multiplicity questions as coefficients of polynomials that relate different symmetric function bases; the latter have been shown to count various combinatorial structures such as tableaux, lattice paths, and tree-like structures. In 1987, Macdonald introduced a new symmetric function basis which profoundly enriched the Theory of Symmetric Functions by its deep connections with Representation Theory, Algebraic Combinatorics, Algebraic Geometry, Particle Physics and Statistics. The connection of Macdonald's basis to Representation Theory was formulated in 1988 by the PI in a program to prove the positivity of the q,t-Kotska polynomials (a truly seminal conjecture in Macdonald's paper). The PI's program was to show that certain modified forms of the Macdonald polynomials are images, under the Frobenius map, of the character of certain bigraded modules; the q,t-Kotska would then yield multiplicities of irreducible representations in the various bi-homogeneous subspaces of these modules. In 1990 the PI and Mark Haiman constructed these bigraded modules as submodules of the Diagonal Harmonics and showed that the q,t-Kotska conjecture would follow if for each partition of n, the module corresponding to it could be shown to have dimension n!. This came to be known as the n!-conjecture, which was proved in 2001 by Mark Haiman after a decade of intensive research in the Algebraic Geometry of Hilbert schemes.The proposed research is to study connections between the Representation Theory of Diagonal Harmonics, the Theory of Macdonald Polynomials and the so-called Parking Functions of Computer Science. The program is a three-pronged attack on the 2002 Shuffle Conjecture, which expresses the Frobenius image of the Character of Diagonal Harmonics as a weighted sum of Parking Functions. The PI plans to work on the Representation Theory parts of the project and is guiding his PhD students to carry out the Symmetric Function Theory and Combinatorial parts of the subject. In the decades that followed the Macdonald paper the PI, his collaborators, and his students have obtained a vast collection of results in the theory of Macdonald polynomials which have already yielded proofs of various special cases of the Shuffle conjecture. Whatever the outcome of the present project, as is common when working on difficult mathematical problems, the proposed work will lead to discoveries that may even surpass in significance the solution of the original problem. The research to be carried out under this grant involves transfer of information from pure algebraic constructs to explicit symmetric functions and ultimately to combinatorial objects such as tableaux, paths and trees. The algebraic problems proposed can only be solved by advances in the theory of symmetric functions. Symmetric functions, in turn, are a computational device of wide applicability. Thus progress in the proposed research should widen the variety of tools available to physicists and engineers for obtaining the hard data needed in their pursuit of applications of science. All the proposed research lies in areas of Mathematics that computers have transformed into experimental sciences. In this setting even beginning students can experience the joy of non-trivial discovery. Thus, this work is an ideal setting in which to convey to our new generations of researchers a deeper understanding of the wide range of possibilities offered by computer-guided research.