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Abstract: A quintessential problem in the theory of symmetric functions is to prove that a given function is symmetric and Schur positive. Better still, one hopes that such a proof will provide a combinatorial interpretation for the Schur coefficients. For example, if the function arises as the (possibly graded) character of a representation of the symmetric group or general linear group, then the Schur expansion gives the irreducible decomposition of the representation. In this talk, we present a general framework for giving a combinatorial proof of symmetry and Schur positivity of an arbitrary function expressed as a sum of quasi-symmetric functions. This method uses a combinatorial construction called dual equivalence graphs (or, more generally, D graphs), which in some ways act as the symmetric group analog of crystal graphs for the general linear group. We will outline the construction of D graphs for several classes of functions, including Macdonald polynomials, and point to some connections with representation theory.
This talk will be accessible to undergraduates.
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