Linear Sections of the Generic Square Hankel Matrix

Hankel matrices (or their Toeplitz versions) have a classic history in Analysis and Functional Operator Theory. One of the best-known problems, which permeates the theory of orthogonal polynomials, is the Hamburger Momentum Problem. Some modern aspects of this activity are reflected in probabilities. Typical Hankel matrices in this connection are matrices with real or complex number entries, and some of the problems ask when they enjoy the property of being positive-definite. Furthermore, these matrices are allowed to be infinite. In algebraic geometry a Hankel matrix appears with functional or, more concretely, polynomial entries in an arbitrary number of indeterminates. The best-known model is when these entries are the same as the indeterminate ones. In this form, they give rise to some of the determinantal ideals studied in the book. Two examples of important algebraic varieties that can be defined using Hankel matrices are the secant manifolds of the classic rational normal curve and the normal rational scrolls. The chapter gives an overall survey of the generic square Hankel matrix with a combinatorial flavor that reflects upon the related algebraic invariants. The discussion of Hankel linear sections covers a good mileage in the chapter, with an emphasis on the section now known as a sub-Hankel matrix. For the early commutative algebra treatment of Hankel matrices we refer to Watanabe (Proc School Sci Tokai Univ 32:11–21, 1997) and Conca (Adv Math 138:263–292, 1998).