Finite Group Actions on Free Resolutions

点击翻译按钮获取中文摘要

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.