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年的《美国复苏与再投资法》（公法111-5）资助的。如果多项式戒指承认对称群体的行动，那么自然要考虑交替进行多项式，即在任何换位时都会改变符号的多项式。在自然情况下，多项式和相关空间的家族发现自己是交换代数，代数组合，代数几何，代数几何，表示理论和近似理论的基本对象。 该项目的目的是研究其计算，组合，代数和几何方面。特别是，将考虑Q，T-Catalan数字，Hilbert计划，最少的免费分辨率，乘数理想和跳跃数字。麦克唐纳对称多项式的理论刺激了Garsia，Haiman和合作者的Q，T-Catalan数量的研究。 PI将与Li合作进一步探索它们。该项目的主要对象将是通过两组或多种变量及其各种不变的多项式产生的理想。 多个变量中多项式方程的交换代数研究系统。在多项式方程式中，对称多项式和交流多项式自然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。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受益于量子代数，密码学和编码理论以及上述领域。