Finite Group Actions on Free Resolutions

自由解的有限群动​​作

• 批准号: 2200844
• 负责人: Federico Galetto
• 金额: $ 12.34万
• 依托单位: Cleveland State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200844&HistoricalAwards=false
• 关键词: Finite; Group; Actions; Free; Resolutions

Solving systems of polynomial equations is a central problem in algebra with many practical applications. The solutions of a system can be regarded as a geometric object, called a variety, leading to a fruitful interplay between algebra and geometry. Varieties can be described by means of numerical parameters, such as their dimension, a measure of size, their degree, a measure of complexity, and more generally their so-called Betti numbers, which provide additional information of algebraic and geometric significance. In this project, the PI aims to describe as explicitly as possible the Betti numbers of some well-known systems of polynomial equations by making use of the inherent symmetries of the underlying varieties. The study will rely on computer algorithms previously developed by the PI for this specific purpose. The grant provides support for graduate students, who will be involved in collecting and analyzing of data, thus providing newer generations of scientists with a diverse background an opportunity to experience hands-on research in addition to learning some advanced mathematics.The goal of this project is to describe the minimal free resolutions of two families of ideals in polynomial rings stable under finite group actions. The first family contains toric ideals of complete graphs and is relevant in the context of combinatorial commutative algebra. The second family contains De Concini-Procesi ideals, which arise in algebraic topology and geometric representation theory. These two families of ideals have, for various reasons, generated significant interest in the literature, and share some traits that will make this research program more relevant. First, their minimal free resolutions and Betti numbers are not fully understood, and this project will make use of novel computational techniques along with representation theory to further current understanding. Second, both are families of non-monomial ideals; this is significant because current techniques for minimal free resolutions with finite group actions are limited to monomial ideals with the action of a symmetric group permuting the variables. In summary, this project will provide new insight into specific families of ideals, while offering an opportunity to develop more widely applicable techniques for constructing resolutions with finite group actions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在多个实际应用的代数中，多项式方程的求解系统是代数的核心问题。系统的溶液可以被视为一种几何对象，称为一种变化，从而导致代数和几何形状之间的富有成果的相互作用。可以通过数值参数来描述品种，例如它们的尺寸，大小的度量，其程度，复杂性度量以及更普遍的所谓贝蒂数字，这些数字提供了代数和几何意义的其他信息。在该项目中，PI旨在通过利用基础品种的固有对称性来显式地描述一些多项式方程系统的Betti数字。该研究将依靠PI先前为此特定目的开发的计算机算法。该赠款为研究生提供了支持，他们将参与收集和分析数据，从而为具有多样化背景的新世代提供了一个机会，这是一个体验动手研究的机会，除了学习一些高级数学外，该项目的目的是描述在有限的小组中稳定的多项式环中的两个理想家庭的最低自由决议。第一个家庭包含完整图的复合理想，并且在组合交换代数的背景下具有相关性。第二个家庭包含de concini-procesi理想，这些理想是代数拓扑和几何表示理论。由于各种原因，这两个理想家族对文献产生了重大兴趣，并具有一些使该研究计划更相关的特征。首先，他们最少的自由决议和贝蒂数字尚未完全理解，该项目将利用新颖的计算技术以及代表理论来进一步的当前理解。其次，两者都是非公主理想的家族。这很重要，因为具有有限群体作用的最小自由分辨率的当前技术仅限于单一理想，而对称组的作用将变量定为变量。总而言之，该项目将为特定的理想家庭提供新的见解，同时为通过有限的小组行动提供了更广泛适用的技术来开发更广泛适用的技术。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。