Generalizations of Schur functions

Schur 函数的推广

• 批准号: RGPIN-2015-03915
• 负责人: vanWilligenburg, Stephanie
• 金额: $ 1.46万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667558
• 关键词: Generalizations; Schur; functions

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''.**

库奇（Cauchy）在1815年首先研究了Schur功能，尽管它们是以Schur的名字命名的，后者在1901年表明它们与Frobenius角色图下的对称群体的不可还原性具有同构。从那时起，它们就在各个区域中出现，包括代数几何形状，Schur函数与复杂的司司曼尼亚人的舒伯特类别的类别一致，以及与量子状态相关的量子力学。******它们也形成了对称功能的Hopf代数的基础。该代数是甲状不断函数的Hopf代数的子代数，其函数同样无处不在，在许多义中出现，包括在对称组上的某个分布方面的概率，同时是组合型Hopf Algebras的终端对象。准对称Schur函数。这些是调查的关键功能，因为有关此类功能的知识将立即影响上述所有领域。我自己发现了这样的功能，我的总体目标是进一步研究这些功能，然后将这些新发现的知识应用于众所周知的开放问题。这将给出代数的几何解释对准对象SCHUR函数，从而推广了前面描述的Schur函数。******作为一种应用，我将确定一个组合规则，以表达谎言表示为列层schur函数的总和。然后，由于Schur函数与准对象SCHUR函数之间的亲密关系，此结果将立即产生影响，解决了代表理论中长期存在的开放问题，以找到一个组合规则来表达谎言代表作为Schur函数的总和。这些改进将为攻击长期存在的开放问题提供新的工具，例如为舒伯特多项式找到产品规则，并解决麦克唐纳多项式猜想，称为``科幻小说''。**