Symmetric function character bases

从19世纪末到20世纪初，表示论先驱们的数学构成了组合表示论的基本工具箱。在那个时代考虑的一些问题今天仍然是代数组合学中的开放问题。 舒尔函数是典型的例子类型的工具，用于回答几何和表示理论的问题。 它们既是对称群的不可约特征标的Frobenius象，又是不可约Gl_n表示的特征标（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编码了对称和一般线性群的表示论的组合数学。 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S_n表示的张量的问题（这可以转化为两个Schur函数的Kronecker积）。 用于理解这个问题和计算分解的数学已经存在了100年，一个组合规则，给了我们一些关于这些系数的直觉，类似于两个不可约Gl_n模的张量的计算（Littlewood-Richardson规则）还没有被发现。 在这个问题上取得如此小的进展之后，一些研究人员说，这表明这条规则并不存在。 在Rosa Orellana最近的一篇论文中，我们引入了对称函数的非齐次基，这些对称函数是被认为是置换矩阵子群的对称群的特征。 这是当函数的变量被专门化为置换矩阵的特征值时，其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标，就像Schur函数是不可约Gl_n模的特征标一样。 这是一个新的范例，因为对称群的特征编码列严格多集表的组合，就像舒尔函数编码列严格表的组合一样，而且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。 它提供了一个新的组合模型，我们可以编码表示理论数据。

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.