Commutative Algebra of Alternating Polynomials

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。如果多项式环允许对称的群体诉讼，自然会考虑交替多项式，即当任何换位作用于其上时改变符号的多项式。在自然情况下，交错多项式及其相关空间族已成为交换代数、代数组合学、代数几何、表示理论和逼近理论中的基本对象。这个项目的目标是研究它们的计算、组合、代数和几何方面。特别是，q，t-Catalan数，Hilbert格式，最小自由分解，乘子理想和跳跃数将被考虑。由Garsia、Haiman等人提出的q，t-Catalan数的研究，受到了Macdonald对称多项式理论的启发。PI将与李合作进一步探索这些问题。这个项目的主要目标将是由两组或更多组变量的交错多项式产生的理想，以及它们的各种不变量。交换代数研究多变量多项式方程组。在多项式方程中，对称多项式和交错多项式自然而然地出现在包括组合学、表示论和粒子物理在内的许多科学分支中。对它们的系统和解决方案的研究将产生新的猜想和定理，这可能会对量子代数、密码学和编码理论以及上述领域有所裨益。

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.