Representation Theoretical Methods in the Theory of Special Functions

拟议的研究扩展到表示理论，对称函数理论和组合数学。 Frobenius映射提供了表示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，lattice path和树形结构。在1987年，麦克唐纳介绍了一个新的对称函数的基础，深刻丰富了理论的对称函数的深刻联系表示论，代数组合学，代数几何，粒子物理和统计。麦克唐纳基与表示论的联系在1988年由PI在一个程序中阐述，以证明q，t-Kotska多项式的正性（麦克唐纳论文中的一个真正开创性的猜想）。 PI的计划是证明麦克唐纳多项式的某些修改形式是某些双阶模的特征在Frobenius映射下的图像; q，t-Kotska将在这些模的各种双齐次子空间中产生不可约表示的多重性。 在1990年，PI和Mark Haiman构造了这些双阶模作为对角调和函数的子模，并证明了q，t-Kotska猜想，如果对于n的每个划分，对应的模可以被证明具有维数n！这被称为n！猜想，这是在2001年由Mark Haiman经过十年的深入研究证明了代数几何的希尔伯特schemes.拟议的研究是研究对角调和表示理论，麦克唐纳多项式理论和所谓的停车函数的计算机科学之间的联系. 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图像表示为Parking函数的加权和。 PI计划在该项目的表示论部分工作，并指导他的博士生进行对称函数理论和组合部分的主题。 在几十年后，麦克唐纳文件的PI，他的合作者，和他的学生已经获得了大量的收集结果的理论麦克唐纳多项式已经产生的证明各种特殊情况下的洗牌猜想。无论当前项目的结果如何，正如在解决困难的数学问题时所常见的那样，所提出的工作将导致发现，甚至可能在意义上超过原始问题的解决方案。 根据这项资助进行的研究涉及将信息从纯代数结构转移到显式对称函数，并最终转移到组合对象，如tableaux，路径和树。提出的代数问题只能通过对称函数理论的进步来解决。对称函数又是一种具有广泛适用性的计算工具。 因此，拟议研究的进展应该会拓宽物理学家和工程师可用的工具种类，以获取他们追求科学应用所需的硬数据。所有被提议的研究都在计算机已经转变为实验科学的数学领域。 在这种情况下，即使是初学者也可以体验到非平凡发现的乐趣。 因此，这项工作是一个理想的设置，其中传达给我们的新一代研究人员更深入地了解计算机引导的研究所提供的广泛的可能性。

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.