Symmetric functions in Combinatorics and Representation Theory

PI将使用对称函数理论中的新工具解决组合学和表示论中的重要问题。拟议的计划将发展理论的对偶等价图，一个强大的，组合的工具，建立Schur积极的拟对称函数，并提供新的应用程序的重要类别的功能。该计划将深化对偶等价图和晶体图之间的联系，同时将对偶等价图推广到其他经典类型，从而扩展对偶等价图机器对经典群表示理论的适用性。该计划还将介绍一个新的家庭的对称函数，这将提供一个组合的理解稳定的克罗内克系数，这反过来，应该导致更好地理解克罗内克系数。对称函数理论在许多数学领域，包括代数组合学，表示论，李群和李代数，代数几何和特殊函数的理论起着重要的作用。这一领域的成果可以应用于数学的许多其他领域以及物理学和计算机科学。除了解决对称函数中重要的长期存在的问题外，该计划还将开发多功能的新工具，以解决基本的开放问题并加深我们的理解。

The PI will address important problems in combinatorics and representation theory using new tools in symmetric function theory. The proposed program will develop the theory of dual equivalence graphs, a powerful, combinatorial tool for establishing the Schur positivity of quasisymmetric functions, and provide new applications to important classes of functions. The program will deepen the connection between dual equivalence graphs and crystal graphs while generalizing dual equivalence graphs to other classicaltypes, thereby expanding the applicability of the dual equivalence graph machinery to representation theory of classical groups. The program will also introduce a new family of symmetric functions that will provide a combinatorial understanding of the stable Kronecker coefficients, which, in turn, should lead to a better understanding of the Kronecker coefficients.Symmetric function theory plays an important role in many areas of mathematics including algebraic combinatorics, representation theory, Lie groups and Lie algebras, algebraic geometry and the theory of special functions. Results in this area have applications to many other areas of mathematics as well as to physics and computer science. In addition to solving important, long-standing problems in symmetric functions, the proposed program will develop versatile new tools with which to tackle fundamental open problems and deepen our understanding.