Spaces of tensors and their syzygies via representation theory and combinatorics

PI将研究一些关于承认一般线性群的乘积作用的代数变量的方程和合子的开放问题。为了研究在专门化方面表现良好的问题，所采用的主要技术源于经典的极化方法，该方法用无平方单项式代替任意单项式。这些技术涉及到PI的论文以及其他一些作者最近的作品中发展出来的一系列新思想，这些思想使用Schur-Weyl对偶性来将关于方程/坐标环/协同的问题转化为一个“通用设置”，在这个“通用设置”中，组合学、拓扑学和对称群的表示理论的工具随时可用。PI将研究一些经典变量的坐标环和协同，以期应用于组合学中的体积问题，或理解代数几何中射影变量的协同的一般渐近结构。本研究属于代数几何领域，主要研究由多项式方程组定义的代数变量。代数变量的Syzygies是更好地理解方程以及变量本身的工具。它们被归纳地定义，按照复杂度的递增顺序，作为方程之间的关系，这些关系之间的关系，等等。当代数变体具有大量的对称性时（这在自然界中是常有的情况），合子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、交换代数和表示理论等方法的组合，以研究张量自然空间（也称为多维矩阵或数组）的方程和协同。我们要解决的一个基本问题是关于线性代数中熟悉的命题的推广，即秩小于r的矩阵是由它们的r × r子矩阵的行列式的消失来定义的。对于大于2维的多维数组，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，信号处理等方面具有重要的应用。

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.