Symmetric function character bases

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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.