Commutative Algebra of Alternating Polynomials

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。如果一个多项式环允许一个对称群作用，那么很自然地要考虑交替多项式，也就是说，当被任何转置作用时，多项式会改变符号。在自然情况下，交错多项式族和相关的空间发现自己是交换代数、代数组合学、代数几何、表示论和逼近论中的基本对象。 这个项目的目标是调查他们的计算，组合，代数和几何方面。特别是，q，t-Catalan数，希尔伯特计划，最小自由决议，乘子理想和跳跃数将被考虑。q，t-Catalan数的研究是由Garsia，Haiman及其合作者引入的，Macdonald对称多项式理论促进了q，t-Catalan数的研究。PI将与Li合作进一步探索它们。这个项目的主要目标是由两组或多组变量的交替多项式产生的理想，以及它们的各种不变量。 交换代数研究多元多项式方程组。在多项式方程中，对称多项式和交替多项式自然出现在许多科学分支中，包括组合学，表示论和粒子物理学。他们的系统和解决方案的项目将导致新的结构和定理，这可能有利于量子代数，密码学和编码理论以及上述领域。

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.