Spaces of tensors and their syzygies via representation theory and combinatorics

通过表示论和组合学研究张量空间及其协同性

• 批准号: 1458715
• 负责人: Claudiu Raicu
• 金额: $ 10.29万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1458715&HistoricalAwards=false
• 关键词: Spaces; tensors; their; syzygies; via

The PI will investigate a number of open problems concerning the equations and syzygies of algebraic varieties admitting the action of a product of general linear groups. The main techniques to be employed stem from the classical method of polarization which replaces arbitrary monomials with square-free monomials in order to study problems that behave well with respect to specialization. The techniques involve a circle of new ideas developed in the PI's thesis as well as in recent works of a number of other authors, which use Schur-Weyl duality in order to translate questions about equations/coordinate rings/syzygies into a "generic setting" where tools from combinatorics, topology and the representation theory of symmetric groups are readily available. The PI will study the coordinate rings and syzygies of a number of classical varieties, with a view towards applications to plethysm problems in combinatorics, or to the understanding of the general asymptotic structure of syzygies of projective varieties in algebraic geometry.The proposed research pertains to the field of algebraic geometry, which is concerned with the study of algebraic varieties defined by systems of polynomial equations. Syzygies of algebraic varieties are a tool to better understand the equations, as well as the varieties themselves. They are defined inductively, in increasing order of their complexity, as the relations between the equations, the relations between these relations, and so on. When the algebraic varieties come equipped with a large group of symmetries, which is often the case in nature, the syzygies will inherit some of those symmetries. The PI and his collaborators will employ a combination of methods from algebraic geometry, combinatorial topology, commutative algebra, and representation theory in order to study the equations and syzygies of natural spaces of tensors (also known as multidimensional matrices or arrays). A fundamental problem that will be addressed concerns the generalization of the familiar statement from linear algebra which states that matrices of rank less than r are defined by the vanishing of the determinants of their r x r submatrices. For multidimensional arrays of dimension larger than two a general statement like this one is unknown, and would have important applications to algebraic statistics, biology, complexity theory, signal processing etc.

PI将调查有关代数品种的方程和共融的许多开放问题，这些方程和共同种类承认了一般线性群体的作用。所用的主要技术源于经典的极化方法，该方法用无方形的单体替代了任意单元，以便研究对专业方面表现良好的问题。这些技术涉及PI论文中发展的一系列新想法，以及在许多其他作者的最新作品中，它们使用Schur-weyl二元性来将有关方程/坐标环/syzygies的问题转化为“通用环境”，其中的工具可以从中使用组合，拓扑和对称组的组合理论。 PI将研究多种经典品种的坐标环和串联，以期待对组合学中的多个问题的应用，或者对代数几何形状中投射品种的普遍渐近性结构的理解，对代数的几何形状的渐近性结构。多项式方程。代数品种的共同体是更好地了解方程式以及品种本身的工具。它们被归纳以越来越复杂的顺序定义，因为方程，这些关系之间的关系等等。当代数品种配备了一大批对称性时，在本质上通常是这种情况时，Syzygies将继承其中一些对称性。 PI和他的合作者将采用代数几何形状，组合拓扑，交换代数和代表理论的方法组合，以研究张量的自然空间的方程式和综合（也称为多维矩阵或阵列）。将解决的一个基本问题涉及线性代数的熟悉陈述的概括，该陈述指出，等级的矩阵小于R的矩阵是由于其R X r子膜的决定因素消失而定义的。对于大于两个尺寸的多维阵列，像这样的一般性陈述是未知的，并且对代数统计，生物学，复杂性理论，信号处理等具有重要的应用。