Spaces of tensors and their syzygies via representation theory and combinatorics

PI将调查一些开放的问题有关的方程和syzygies代数品种承认行动的产品一般线性群。要采用的主要技术源于经典的极化方法，该方法用无平方根的单项式代替任意单项式，以研究在专业化方面表现良好的问题。该技术涉及一圈新的想法发展中的PI的论文，以及在最近的作品中的一些其他作者，其中使用舒尔-外尔对偶，以翻译问题的方程/坐标环/syzygies到一个“通用设置”的工具，从组合，拓扑和表示理论的对称群是现成的。PI将研究坐标环和syzygies的一些经典的品种，以期对应用程序的体积问题在组合，或一般渐近结构的理解的syzygies的投影簇在代数几何。拟议的研究属于代数几何领域，这是关注的研究代数簇定义的多项式方程组。代数簇的合势是更好地理解方程和簇本身的工具。它们被归纳地定义为方程之间的关系，这些关系之间的关系，等等，当代数簇具有一大群对称性时，这在自然界中是经常发生的，合冲将继承其中的一些对称性。PI和他的合作者将采用代数几何，组合拓扑，交换代数和表示论的组合方法，以研究张量（也称为多维矩阵或数组）的自然空间的方程和合势。一个基本的问题，将解决有关的推广熟悉的声明，从线性代数的国家，矩阵的秩小于r的定义消失的决定因素，他们的r x r的子矩阵。对于维数大于2的多维数组，像这样的一般陈述是未知的，并且在代数统计，生物学，复杂性理论，信号处理等方面有重要的应用。

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.