Symmetric function character bases

对称函数字符库

• 批准号: RGPIN-2017-05724
• 负责人: Zabrocki, Mike
• 金额: $ 1.17万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711780
• 关键词: Symmetric; function; character; bases

The mathematics of the pioneers of representation theory from the end of the 19th century to the beginning of the 20th century make up the basic toolbox in combinatorial representation theory. Some of the questions that were considered in that era are still around today as motivating open questions in algebraic combinatorics. Schur functions are the typical example of the type of tools that are used in answering geometric and representation theoretic questions. They are both the Frobenius image of the irreducible characters of the symmetric group and the characters of an irreducible Gl_n representation (as a function of the eigenvalues of the matrix). These functions simultaneously encode combinatorics of the representation theory of the symmetric and general linear groups through structure coefficients and change of basis coefficients. They allow us to apply linear algebra to answer questions in other mathematical areas. Consider however the question of decomposing the tensor of two irreducible S_n representations (this can be translated to the Kronecker product of two Schur functions). The mathematics for understanding this problem and computing the decomposition has been around for 100 years, a combinatorial rule that gives us some intuition about these coefficients similar to the computation of the tensor of two irreducible Gl_n modules (the Littlewood-Richardson rule) has not yet been discovered. After such little progress that has been made on this problem, some researchers say that this indicates that this rule doesn't exist. In a recent paper with Rosa Orellana, we introduce an in-homogeneous basis of the symmetric functions that are the characters of the symmetric group considered as the subgroup of permutation matrices. This is a basis that when the variables of the functions are specialized to the eigenvalues of a permutation matrix, the values are symmetric group characters. The elements of this basis are the characters of the irreducible symmetric group representations in the same way that the Schur functions are the characters of the irreducible Gl_n modules. This is a new paradigm because the characters of the symmetric group encode the combinatorics of column strict multi-set tableaux in the same way that Schur functions encode the combinatorics of column strict tableaux and this combinatorial object of multi-sets and multi-set tableaux does not seem to appear in the literature on symmetric group representation theory. It provides a new combinatorial model by which we can encode representation theoretical data.

19世纪末20世纪初表示论先驱的数学思想构成了组合表示论的基本工具箱。在那个时代被考虑的一些问题今天仍然存在，因为它们激发了代数组合学中的开放问题。 舒尔函数是用于回答几何和表示理论问题的工具的典型例子。它们既是对称群的不可约特征标的Frobenius映象，也是不可约Gl_n表示的特征标(作为矩阵本征值的函数)。这些函数通过结构系数和基系数的变化同时对对称和一般线性群的表示理论的组合进行编码。它们允许我们应用线性代数来回答其他数学领域的问题。 然而，考虑分解两个不可约的S_n表示的张量的问题(这可以转化为两个Schur函数的Kronecker积)。理解这个问题和计算分解的数学已经存在了100年，但还没有发现一种组合规则，它可以给我们一些直观的关于这些系数的计算，类似于计算两个不可约Gl_n模的张量(Littlewood-Richardson规则)。在这个问题上取得了如此小的进展后，一些研究人员表示，这表明这一规则不存在。 在最近与Rosa Orellana的一篇文章中，我们引入了对称函数的非齐次基，它是被认为是置换矩阵子群的对称群的特征。这是当函数的变量专用于置换矩阵的特征值时，值是对称群字符的基础。这个基的元素是不可约对称群表示的特征，正如Schur函数是不可约Gl_n模的特征。 这是一种新的范式，因为对称群的特征标以与Schur函数编码列严格多集表的组合相同的方式编码列严格多集表的组合，而这种多集和多集表的组合对象似乎没有出现在对称群表示理论的文献中。它提供了一种新的组合模型，可以用来对表示理论数据进行编码。