Commutative Algebra of Alternating Polynomials

交替多项式的交换代数

• 批准号: 0901367
• 负责人: Bernd Ulrich
• 金额: $ 9.12万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901367&HistoricalAwards=false
• 关键词: Commutative; Algebra; Alternating; Polynomials

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.

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对称群作用，那么很自然地考虑交替多项式，即，当被任何换位作用时改变符号的多项式。在自然情况下，交替多项式族和相关空间已经发现自己是交换代数、代数组合学、代数几何、表示理论和近似理论中的基本对象。这个项目的目标是研究它们的计算、组合、代数和几何方面。特别地，q，t-加泰罗尼亚数，希尔伯特格式，最小自由分辨率，乘数理想和跳跃数将被考虑。由Garsia， Haiman及其合作者引入的q，t-加泰罗尼亚数的研究受到麦克唐纳对称多项式理论的刺激。PI将与li合作进一步探索。这个项目的主要目标将是由交替多项式在两组或多组变量中产生的理想，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，对称多项式和交替多项式自然地出现在许多科学分支中，包括组合学、表示理论和粒子物理学。关于他们的系统和解决方案的项目将导致新的猜想和定理，这些猜想和定理可能有益于量子代数、密码学和编码理论以及上述领域。