Generalizations of Schur functions

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Schur functions were first studied by Cauchy in 1815, although they were named after Schur, who in 1901 showed that they were isomorphic to an irreducible character of a symmetric group under the Frobenius character map. Since then they have arisen in a variety of areas including algebraic geometry where Schur functions agree with Schubert classes in the cohomology ring of the complex Grassmannian, and quantum mechanics where they are related to quantum states.******They also form a basis of the Hopf algebra of symmetric functions. This algebra is a subalgebra of the Hopf algebra of quasisymmetric functions, whose functions are equally ubiquitous, arising in many guises including as probabilities with respect to a certain distribution on the symmetric groups, and together being the terminal object in the category of combinatorial Hopf algebras.******Therefore natural functions to study are quasisymmetric refinements of Schur functions, that is, quasisymmetric Schur functions. These are key functions to investigate as knowledge about such functions would immediately impact all of the aforementioned areas. Such functions were discovered by myself, Haglund, Luoto and Mason, and my overarching goal is to investigate these functions further and then to apply this new-found knowledge to well-known open problems.******For example, further properties I intend to investigate include the existence of a geometric Littlewood-Richarsdon rule for skew quasisymmetric Schur functions. This would give an algebraic geometric interpretation to quasisymmetric Schur functions, generalizing that of Schur functions described earlier.******As one application, I will determine a combinatorial rule to express Lie representations as a sum of quasisymmetric Schur functions. Then due to the intimate relationship between Schur functions and quasisymmetric Schur functions this result would have immediate impact, resolving the long-standing open problem in representation theory to find a combinatorial rule to express Lie representations as a sum of Schur functions.******Another avenue I intend to investigate is whether generalized Schur functions such as Schubert polynomials and Macdonald polynomials exhibit natural quasisymmetric refinements. These refinements would provide new tools for attacking long-standing open problems such as finding a product rule for Schubert polynomials and resolving the Macdonald polynomial conjectures known as ``Science Fiction''.**