Commutative Algebra of Alternating Polynomials

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).If a polynomial ring admits a symmetric group action, it is natural to consider alternating polynomials, that is, polynomials which change sign when acted on by any transposition. In natural situations, the families of alternating polynomials and related spaces have found themselves to be fundamental objects in commutative algebra, algebraic combinatorics, algebraic geometry, representation theory, and approximation theory. The goal of thisproject is to investigate their computational, combinatorial, algebraic, and geometric aspects. In particular, q,t-Catalan numbers, Hilbert schemes, minimal free resolutions, multiplier ideals and jumping numbers will be considered. The study of q,t-Catalan numbers, which were introduced by Garsia, Haiman and collaborators, has been stimulated by the theory of Macdonald symmetric polynomials. The PI will explore them further in collaboration withLi. The main object of this project will be the ideals generated by alternating polynomials in two or more sets of variables, and their various invariants. Commutative algebra studies systems of polynomial equations in many variables. Among polynomial equations, symmetric polynomials and alternating polynomials naturally occur in many branches of science including combinatorics, representation theory, and particle physics. The project on their systems and solutions will lead to new conjectures and theorems which may benefit quantum algebra, cryptography, and coding theory as well as the areas mentioned above.