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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759186
• 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世纪末至20世纪初表示论先驱的数学构成了组合表示论的基本工具箱。那个时代考虑的一些问题至今仍然存在，作为代数组合学中的开放性问题。舒尔函数是用于回答几何和表示理论问题的工具类型的典型示例。 它们都是对称群的不可约特征的 Frobenius 图像和不可约 Gl_n 表示的特征（作为矩阵特征值的函数）。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论的组合进行编码。 它们使我们能够应用线性代数来回答其他数学领域的问题。然而，请考虑分解两个不可约 S_n 表示的张量的问题（这可以转换为两个 Schur 函数的克罗内克乘积）。 理解这个问题和计算分解的数学已经存在了 100 年，组合规则可以让我们对这些系数有一些直觉，类似于计算两个不可约 Gl_n 模的张量（Littlewood-Richardson 规则）。被发现了。 在这个问题上取得如此小的进展后，一些研究人员表示，这表明这个规则不存在。在最近与 Rosa Orellana 合作的一篇论文中，我们引入了对称函数的非齐次基，该函数的特征是对称群被视为置换矩阵的子群。 这是当函数的变量专门化为置换矩阵的特征值时，这些值是对称群特征的基础。 这个基的元素是不可约对称群表示的特征，就像 Schur 函数是不可约 Gl_n 模的特征一样。这是一个新的范式，因为对称群的特征编码了列严格多的组合数学-set tableaux 的方式与 Schur 函数编码列严格 tableaux 的组合学相同，并且多集和多集 tableaux 的这种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，我们可以通过该模型对表示理论数据进行编码。