Finite Group Actions on Free Resolutions

多项式方程组的求解是代数中的一个核心问题，有着广泛的实际应用。一个系统的解可以被视为一个几何对象，称为簇，它导致了代数和几何之间卓有成效的相互作用。变种可以通过数字参数来描述，例如它们的尺寸、大小的度量、它们的程度、复杂性的度量，以及更一般地它们的所谓的Betti数，它们提供了具有代数和几何意义的附加信息。在这个项目中，PI的目的是通过利用基本变元的内在对称性来尽可能明确地描述一些著名的多项式方程组的Betti数。这项研究将依赖于PI以前为这一特定目的开发的计算机算法。这项资助为研究生提供支持，他们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实践研究的机会，并学习一些高等数学。本项目的目标是描述在有限群作用下稳定的多项式环中两族理想族的最小自由分解。第一族包含完全图的环理想，并且在组合交换代数的上下文中是相关的。第二类包含了代数拓扑学和几何表示论中的de Concini-Procesi理想。由于各种原因，这两个理想家族在文献中引起了极大的兴趣，并分享了一些特征，将使这一研究计划更具相关性。首先，它们的最小自由分辨率和Betti数还没有完全被理解，这个项目将利用新的计算技术和表示理论来进一步理解当前的理解。其次，两者都是非一项式理想族；这一点很重要，因为目前用于具有有限群作用的最小自由分解的技术仅限于具有对称群置换变量作用的单项式理想。总而言之，这个项目将提供对特定理想家族的新见解，同时提供一个机会来开发更广泛适用的技术来构建具有有限群体行动的决议。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.