Spaces of tensors and their syzygies via representation theory and combinatorics

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

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.