Spaces of tensors and their syzygies via representation theory and combinatorics

PI将研究一些公开的问题，涉及允许一般线性群的乘积的作用的代数簇的方程和合集。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代替任意单项，以便研究在专业化方面表现良好的问题。这些技巧涉及PI的论文以及其他一些作者最近的作品中发展的一系列新思想，这些作者使用Schur-Weyl对偶性将关于方程/坐标环/合子的问题转化为“一般环境”，其中来自组合学、拓扑学和对称群表示理论的工具随时可用。PI将研究一些经典簇的坐标环和合子，以期应用于组合数学中的倍数问题，或理解代数几何中射影簇的合子的一般渐近结构。建议的研究属于代数几何领域，涉及由多项式方程组定义的代数簇的研究。代数簇的合系是更好地理解方程以及簇本身的工具。它们按照其复杂性的递增顺序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。当代数变体配备了一大组对称时，合子将继承其中的一些对称，这在自然界中通常是这样的。PI和他的合作者将使用代数几何、组合拓扑学、交换代数和表示论的方法来研究张量(也称为多维矩阵或阵列)的自然空间的方程和合子。要解决的一个基本问题涉及到线性代数中常见的陈述的推广，该陈述指出，排名小于r的矩阵由其r×r个子矩阵的行列式的消失来定义。对于维度大于2的多维数组，这样的一般陈述是未知的，并且将在代数统计、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.