Generalizations of Schur functions

Schur函数最早是由Cauchy在1815年研究的，尽管它们是以Schur的名字命名的，Schur在1901年证明了它们在Frobenius特征标映射下同构于对称群的不可约特征标。从那时起，它们出现在各种领域，包括代数几何，其中Schur函数与复Grassman的上同调环中的Schubert类一致，以及量子力学，其中它们与量子态有关。 它们也构成对称函数的Hopf代数的基。这个代数是拟对称函数的Hopf代数的一个子代数，它的函数同样无处不在，以多种形式出现，包括关于对称群上的某个分布的概率，并且一起是组合Hopf代数范畴的终端对象。 因此，要研究的自然函数是Schur函数的拟对称加细，即拟对称Schur函数。这些是需要调查的关键职能，因为对这些职能的了解将立即影响到上述所有领域。这些函数是由我、Haglund、Luoto和Mason发现的，我的首要目标是进一步研究这些函数，然后将这些新发现的知识应用于众所周知的公开问题。 例如，我打算研究的其他性质包括斜拟对称Schur函数的几何Littlewood-Richarsdon规则的存在性。这将给出准对称Schur函数的代数几何解释，推广了前面描述的Schur函数的几何解释。 作为一个应用，我将确定一个组合规则，将Lie表示表示为准对称Schur函数的和。然后，由于Schur函数和拟对称Schur函数之间的密切关系，这一结果将产生立竿见影的效果，解决了表示论中长期悬而未决的问题，即找到一个组合规则来将Lie表示表示为Schur函数的和。 我打算调查的另一个途径是，舒伯特多项式和麦克唐纳多项式等广义舒尔函数是否表现出自然的准对称精化。这些改进将为解决长期悬而未决的问题提供新的工具，例如寻找舒伯特多项式的乘积规则，并解决被称为“科幻小说”的麦克唐纳多项式猜想。

Schur functions were first studied by Cauchy in 1815, although they were named after Schur, who in 1901 showed that they were isomorphic to an irreducible character of a symmetric group under the Frobenius character map. Since then they have arisen in a variety of areas including algebraic geometry where Schur functions agree with Schubert classes in the cohomology ring of the complex Grassmannian, and quantum mechanics where they are related to quantum states. They also form a basis of the Hopf algebra of symmetric functions. This algebra is a subalgebra of the Hopf algebra of quasisymmetric functions, whose functions are equally ubiquitous, arising in many guises including as probabilities with respect to a certain distribution on the symmetric groups, and together being the terminal object in the category of combinatorial Hopf algebras. Therefore natural functions to study are quasisymmetric refinements of Schur functions, that is, quasisymmetric Schur functions. These are key functions to investigate as knowledge about such functions would immediately impact all of the aforementioned areas. Such functions were discovered by myself, Haglund, Luoto and Mason, and my overarching goal is to investigate these functions further and then to apply this new-found knowledge to well-known open problems. For example, further properties I intend to investigate include the existence of a geometric Littlewood-Richarsdon rule for skew quasisymmetric Schur functions. This would give an algebraic geometric interpretation to quasisymmetric Schur functions, generalizing that of Schur functions described earlier. As one application, I will determine a combinatorial rule to express Lie representations as a sum of quasisymmetric Schur functions. Then due to the intimate relationship between Schur functions and quasisymmetric Schur functions this result would have immediate impact, resolving the long-standing open problem in representation theory to find a combinatorial rule to express Lie representations as a sum of Schur functions. Another avenue I intend to investigate is whether generalized Schur functions such as Schubert polynomials and Macdonald polynomials exhibit natural quasisymmetric refinements. These refinements would provide new tools for attacking long-standing open problems such as finding a product rule for Schubert polynomials and resolving the Macdonald polynomial conjectures known as ``Science Fiction’’.