Combinatorial Structures for Permutation Enumeration and Macdonald Polynomials

摘要：PI计划在三个不同的领域进行研究：对称函数在置换枚举中的应用、麦克唐纳多项式的组合学和白车理论。在置换枚举领域，PI计划扩展Brenti， Remmel, Beck, Langley， Mendes和Wagner的工作，他们已经证明了对称群、高八面体群和环群的环积上的置换统计的许多旧的和新的生成函数可以通过对简单对称函数恒等式应用合适的同态推导出来。本研究项目的目标之一是将同态方法扩展到新的对称函数类和新的对称函数恒等式中。本项目的第二个研究领域是研究麦克唐纳多项式的各种组合方面。例如，在最近的工作中，Haglund， Haiman和Loehr给出了准对称函数中修正麦克唐纳多项式展开中出现的系数的组合解释，并给出了非对称舒尔函数的组合解释。PI计划在Garsia-Haiman模的背景下研究修正Macdonald多项式和相关多项式的拟对称函数展开中出现的系数的代数意义。去年，Mason定义了著名的Robinson-Knuth-Schensted对应的一个修正，它可以用来证明非对称舒尔函数的各种恒等式。哈格伦德、梅森和PI最近的工作导致了这种对应的一个新家族的发现，这使得人们可以对这些对应的许多性质给出统一的证明。私家侦探计划继续研究这些信件。最后，PI还计划研究PI和他的学生B. Miceli开发的新白嘴龙理论模型的应用。这个新的车格理论模型允许人们证明一个相当一般的车格多项式的分解定理，它专门研究了文献中出现的许多这样的分解定理。PI将进一步研究这一新的白嘴鸦理论模型的应用。三个被提议的研究领域中的每一个，对称函数在遍历枚举问题中的应用，麦克唐纳多项式的组合方面，以及新组合模型的车理论，都是目前活跃的研究领域，并与其他数学领域有许多联系。例如，对称函数的组合学在许多数学领域发挥了关键作用，包括多项式方程理论、有限群的表示理论、李代数、代数几何和特殊函数理论。自1988年引入麦克唐纳多项式以来，人们对其进行了深入的研究，并在特殊函数理论、表示理论、代数几何、群论、统计学和量子力学中找到了应用。在这三个领域中，PI正在研究基本组合模型的性质，这些模型允许人们综合大量先前在文献中出现的结果，并证明大量新结果。建议的研究应该导致对在这三个领域中发挥重要作用的基本组合模型的更深入的理解。

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.