Grant: Discovery Grants Program - Individual

The mathematics of the pioneers of representation theory from the end of the 19th century to the beginningof the 20th century make up the basic toolbox in combinatorial representation theory.Some of the questions that were considered inthat era are still around today as motivating openquestions in algebraic combinatorics. Schur functions are the typical exampleof the type of tools that are used in answeringgeometric and representation theoretic questions. They are both the Frobeniusimage of the irreducible charactersof the symmetric group and the charactersof an irreducible Gl_n representation(as a function of the eigenvalues of the matrix).These functions simultaneouslyencode combinatorics of therepresentation theory of the symmetric andgeneral linear groups through structure coefficients andchange of basis coefficients. They allow us to applylinear algebra to answer questions in other mathematical areas. Consider however the question of decomposing the tensor of twoirreducible S_n representations (this can be translated to theKronecker product of two Schur functions). The mathematicsfor understanding this problem andcomputing the decomposition has been around for 100years, a combinatorial rule that gives us some intuitionabout these coefficients similar to the computationof the tensor of two irreducible Gl_n modules(the Littlewood-Richardson rule) has not yet beendiscovered. After such little progress that hasbeen made on this problem, some researcherssay 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 arethe characters of the symmetric group considered as thesubgroup of permutation matrices. This is a basis thatwhen the variables of the functions are specialized to the eigenvalues of a permutationmatrix, the values are symmetric group characters. The elements of this basis are the characters of the irreduciblesymmetric group representations in the same way thatthe Schur functions are the characters of the irreducible Gl_n modules. This is a new paradigm because the characters of thesymmetric group encode the combinatorics of columnstrict multi-set tableaux in the same way that Schurfunctions encode the combinatorics of column strict tableauxand this combinatorial object of multi-sets and multi-settableaux does not seem to appear in the literature onsymmetric group representation theory. It provides a newcombinatorial model by which we can encode representationtheoretical data.