Combinatorial Structures for Permutation Enumeration and Macdonald Polynomials

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Combinatorial Structures for Permutation Enumeration and Macdonald PolynomialsPI: Jeffrey B. Remmel Abstract: The PI plans to pursue research in three different areas: the application of symmetric functions to permutation enumeration, the combinatorics of Macdonald polynomials, and rook theory. In the area of permutation enumeration, the PI plans to extend the work of Brenti, Remmel, Beck, Langley, Mendes and Wagner who have shown that many old and new generating functions for permutation statistics over the symmetric group, hyperoctahedral group, and the wreath products of cyclic groups with the symmetric group can be derived by applying suitable homomorphisms to simple symmetric function identities. One goal of this research project will be to extend the homomorphism method to new classes of symmetric functions and new symmetric function identities. A second area of research in this project is to study various combinatorial aspects of the Macdonald Polynomials. For example, in recent work, Haglund, Haiman, and Loehr gave a combinatorial interpretation of the coefficients that arise in the expansion of the modified Macdonald polynomials in terms of quasisymmetric functions and gave a combinatorial interpretation of non-symmetric Schur functions. The PI plans to study the algebraic meaning of the coefficients that appear in the quasisymmetric function expansion of the modified Macdonald polynomials and related polynomials in the context of Garsia-Haiman modules. Last year, Mason defined a modification of the famed Robinson-Knuth-Schensted correspondence which can used to prove various identities for non-symmetric Schur functions. Recent work of Haglund, Mason, and the PI has lead to the discovery of a new family of such correspondence which allow one to give a uniform proof of many properties of these correspondences. The PI plans to continue the study of these correspondences. Finally the PI also plans to study applications of a new rook theory model developed by the PI and his student B. Miceli. This new rook theory model allows one to prove a quite general factorization theorem for rook polynomials that specializes to many such factorization theorems that have appeared in the literature. The PI will research further applications of this new rook theory model.Each of three proposed research areas, the applications of symmetric functions to permuation enumeration problems, the combinatorial aspects of Macdonald polynomials, and the new combinatorial models for rook theory, are currently active areas of research and have many connections with other areas of mathematics. For example, the combinatorics of symmetric functions have played a key role in many areas of mathematics including the theory of polynomial equations, the representation theory of finite groups, Lie algebras, algebraic geometry and the theory of special functions. Since their introduction in 1988, Macdonald polynomials have been intensely studied and have found applications in special function theory, representation theory, algebraic geometry, group theory, statistics, and quantum mechanics. In each of the three areas, the PI is studying properties of fundamental combinatorial models that allow one to synthesize a large number of results that have previously appeared in the literature as well to prove a large number of new results. The proposed research should lead to a deeper understanding of fundamental combinatorial models which play an important role in each of the three areas.