Finite Group Actions on Free Resolutions

求解多项式方程组是代数学中的一个中心问题，具有许多实际应用。一个系统的解可以被看作是一个几何对象，称为一个簇，导致代数和几何之间富有成效的相互作用。变种可以通过数值参数来描述，例如它们的尺寸，大小的度量，它们的程度，复杂性的度量，以及更一般的所谓贝蒂数，这些数字提供了代数和几何意义的额外信息。在这个项目中，PI的目的是尽可能明确地描述一些著名的多项式方程系统的贝蒂数，通过利用潜在品种的固有对称性。本研究将依赖于PI先前为此特定目的开发的计算机算法。研究生将参与数据的收集和分析，从而为具有不同背景的新一代科学家提供一个机会，除了学习一些高等数学之外，还可以体验动手研究。该项目的目标是描述在有限群作用下稳定的多项式环中两个理想族的最小自由解。第一个家庭包含环面理想的完全图，是相关的上下文中的组合交换代数。第二个家庭包含德Concini-Procesi理想，出现在代数拓扑和几何表示理论。由于各种原因，这两个理想家庭在文献中产生了极大的兴趣，并分享了一些使本研究计划更具相关性的特征。首先，它们的最小自由分辨率和贝蒂数还没有完全理解，这个项目将利用新的计算技术沿着表示理论，以进一步了解当前的。其次，两者都是非单项式理想的家庭，这是重要的，因为目前的技术与有限的组行动的最小自由决议是有限的单项式理想与对称组置换变量的行动。总之，这个项目将提供新的见解，以具体家庭的理想，同时提供了一个机会，以发展更广泛适用的技术，建设决议与有限的群体actions.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.