Combinatorial Structures for Permutation Enumeration and Macdonald Polynomials

排列枚举和麦克唐纳多项式的组合结构

• 批准号: 0654060
• 负责人: Jeffrey Remmel
• 金额: $ 11.35万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654060&HistoricalAwards=false
• 关键词: Combinatorial; Structures; Permutation; Enumeration; Macdonald

Combinatorial Structures for Permutation Enumeration and Macdonald PolynomialsPI: Jeffrey B. Remmel Abstract: The PI plans to pursue research in three different areas: the application of symmetric functions to permutation enumeration, the combinatorics of Macdonald polynomials, and rook theory. In the area of permutation enumeration, the PI plans to extend the work of Brenti, Remmel, Beck, Langley, Mendes and Wagner who have shown that many old and new generating functions for permutation statistics over the symmetric group, hyperoctahedral group, and the wreath products of cyclic groups with the symmetric group can be derived by applying suitable homomorphisms to simple symmetric function identities. One goal of this research project will be to extend the homomorphism method to new classes of symmetric functions and new symmetric function identities. A second area of research in this project is to study various combinatorial aspects of the Macdonald Polynomials. For example, in recent work, Haglund, Haiman, and Loehr gave a combinatorial interpretation of the coefficients that arise in the expansion of the modified Macdonald polynomials in terms of quasisymmetric functions and gave a combinatorial interpretation of non-symmetric Schur functions. The PI plans to study the algebraic meaning of the coefficients that appear in the quasisymmetric function expansion of the modified Macdonald polynomials and related polynomials in the context of Garsia-Haiman modules. Last year, Mason defined a modification of the famed Robinson-Knuth-Schensted correspondence which can used to prove various identities for non-symmetric Schur functions. Recent work of Haglund, Mason, and the PI has lead to the discovery of a new family of such correspondence which allow one to give a uniform proof of many properties of these correspondences. The PI plans to continue the study of these correspondences. Finally the PI also plans to study applications of a new rook theory model developed by the PI and his student B. Miceli. This new rook theory model allows one to prove a quite general factorization theorem for rook polynomials that specializes to many such factorization theorems that have appeared in the literature. The PI will research further applications of this new rook theory model.Each of three proposed research areas, the applications of symmetric functions to permuation enumeration problems, the combinatorial aspects of Macdonald polynomials, and the new combinatorial models for rook theory, are currently active areas of research and have many connections with other areas of mathematics. For example, the combinatorics of symmetric functions have played a key role in many areas of mathematics including the theory of polynomial equations, the representation theory of finite groups, Lie algebras, algebraic geometry and the theory of special functions. Since their introduction in 1988, Macdonald polynomials have been intensely studied and have found applications in special function theory, representation theory, algebraic geometry, group theory, statistics, and quantum mechanics. In each of the three areas, the PI is studying properties of fundamental combinatorial models that allow one to synthesize a large number of results that have previously appeared in the literature as well to prove a large number of new results. The proposed research should lead to a deeper understanding of fundamental combinatorial models which play an important role in each of the three areas.

置换列举和麦克唐纳多项式SPI的组合结构：Jeffrey B. Remmel摘要：PI计划在三个不同领域进行研究：对称函数在置换枚举中的应用，麦当劳多项式的组合和ROOK理论。在列出列举领域，PI计划扩展Brenti，Remmel，Beck，Langley，Mendes和Wagner的工作，他们已经表明，在对称组中，对对称群体，超肠hedral群体以及与对称组的环形组的合适群体的循环构成相对态度的许多旧生成功能可以衍生功能。该研究项目的一个目标是将同态方法扩展到新的对称函数和新的对称功能身份。该项目的第二个研究领域是研究麦当劳多项式的各个组合方面。例如，在最近的工作中，Haglund，Haiman和Loehr对用于准对称功能的修饰的MacDonald多项式扩展时产生的系数进行了组合解释，并给出了非对称SCHUR功能的组合解释。 PI计划研究在Garsia-Haiman模块的背景下，经过修改的Macdonald多项式和相关多项式的准对称函数扩展中出现系数的代数含义。去年，梅森（Mason）定义了著名的鲁滨逊 - 斯诺特 - 施法德对应关系的修改，该通信可用于证明非对称Schur函数的各种身份。 Haglund，Mason和PI的最新工作导致发现了一个新家族，这使人们可以统一证明这些对应关系的许多特性。 PI计划继续研究这些对应关系。最终，PI还计划研究PI和他的学生B. Miceli开发的新的Rook理论模型的应用。这个新的Rook理论模型允许人们证明对Rook多项式的一般分解定理，该定理专门研究文献中出现的许多此类分解定理。 PI将研究这种新的Rook理论模型的进一步应用。每个提出的研究领域，对称功能在次数枚举问题上的应用，麦克唐纳多项式多项式的组合方面以及ROOK理论的新组合模型，目前是研究的积极领域，并且与其他数学领域有许多联系。例如，对称函数的组合在许多数学领域都起着关键作用，包括多项式方程理论，有限群体的代表理论，lie代数，代数几何学和特殊功能理论。自1988年引入以来，麦当劳多项式已经进行了深入研究，并在特殊功能理论，表示理论，代数几何，群体理论，统计和量子力学中找到了应用。在这三个领域中的每个领域中，PI都在研究基本组合模型的特性，这些模型允许人们合成以前在文献中出现的大量结果，以证明大量新的结果。拟议的研究应导致对基本组合模型的更深入了解，这些模型在这三个领域中的每个领域都起着重要作用。