Symmetric functions in Combinatorics and Representation Theory

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The PI will address important problems in combinatorics and representation theory using new tools in symmetric function theory. The proposed program will develop the theory of dual equivalence graphs, a powerful, combinatorial tool for establishing the Schur positivity of quasisymmetric functions, and provide new applications to important classes of functions. The program will deepen the connection between dual equivalence graphs and crystal graphs while generalizing dual equivalence graphs to other classicaltypes, thereby expanding the applicability of the dual equivalence graph machinery to representation theory of classical groups. The program will also introduce a new family of symmetric functions that will provide a combinatorial understanding of the stable Kronecker coefficients, which, in turn, should lead to a better understanding of the Kronecker coefficients.Symmetric function theory plays an important role in many areas of mathematics including algebraic combinatorics, representation theory, Lie groups and Lie algebras, algebraic geometry and the theory of special functions. Results in this area have applications to many other areas of mathematics as well as to physics and computer science. In addition to solving important, long-standing problems in symmetric functions, the proposed program will develop versatile new tools with which to tackle fundamental open problems and deepen our understanding.