Representation Theoretical Methods in the Theory of Special Functions

这项研究涉及表象理论、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，它将维度和重数问题编码为关联不同对称函数基的多项式的系数；后者已被证明计算各种组合结构，如表结构、格子路径和树状结构。1987年，Macdonald引入了一种新的对称函数基，它与表示论、代数组合学、代数几何、粒子物理学和统计学有着深刻的联系，极大地丰富了对称函数论。麦克唐纳的基础与表示理论的联系是在1988年由PI在一个证明q，t-Kotska多项式的正性的程序中提出的(在Macdonald的论文中这是一个真正开创性的猜想)。PI的程序是为了证明在Frobenius映射下，Macdonald多项式的某些修改形式是某些双格化模的特征的映象；然后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重数。1990年，Pi和Mark Haiman构造了这些双格化模作为对角调和的子模，并证明了q，t-Kotska猜想成立，如果对n的每一个分拆，对应的模都有n维！这就是著名的n！猜想，这是Mark Haiman在对Hilbert模式的代数几何进行了十年的深入研究后于2001年证明的。这项研究旨在研究对角调和表示理论、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关系。该程序是对2002年的Shuffle猜想的三管齐下的攻击，该猜想将对角调和的特征的Frobenius映象表示为停车函数的加权和。PI计划在该项目的表示理论部分开展工作，并指导他的博士生完成该学科的对称函数理论和组合部分。在麦克唐纳发表论文后的几十年里，圆周率、他的合作者和他的学生在麦克唐纳多项式理论方面获得了大量的结果，这些结果已经产生了洗牌猜想的各种特殊情况的证明。无论当前项目的结果如何，就像在处理困难的数学问题时常见的那样，拟议的工作将导致甚至在意义上超过原始问题的解决的发现。在这项资助下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步来解决。反过来，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扩大物理学家和工程师可用来获得科学应用所需的硬数据的各种工具。所有拟议的研究都集中在计算机已转化为实验科学的数学领域。在这种背景下，即使是初学者也能体验到喜悦的非同小可的发现。因此，这项工作是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理解的理想环境。

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.