Efficient $k$-Sign Consistency Verification of Hankel Matrices via Schur Polynomials

TL;DR

This paper introduces a simplified method for certifying $k$-sign consistency of Hankel matrices using Schur polynomial theory, reducing computational complexity and establishing necessary and sufficient conditions for Hankel operators.

Contribution

It provides a novel formula expressing Hankel matrix minors as nonnegative integer combinations of consecutive minors, enabling efficient $k$-sign consistency verification.

Findings

The method simplifies $k$-sign consistency verification for Hankel matrices.

The derived formula relates minors to Schur polynomials and Littlewood--Richardson coefficients.

Results extend to Toeplitz matrices and have partial analogues for circulant matrices.

Abstract

We consider the problem of certifying (strict) $k$-sign consistency of a matrix, that is, whether all of its $k$-th order minors share the same (strict) sign. Although this problem is generally of combinatorial complexity, we show that for Hankel matrices it can be significantly simplified: our sufficient condition requires checking only the $k$-th order minors of a reshaped Hankel matrix with $k$ rows. Remarkably, when applied to the Hankel operator, this sufficient condition is also necessary. Comparable results were known only in the setting of (strictly) $k$-positive Hankel matrices and operators, in which all minors of order up to $k$ have the same (strict) sign. More concretely, we derive a formula expressing the $k$-th order minors of Hankel matrices as nonnegative integer linear combinations of $k$-th order minors with consecutive row indices. Our derivation uses Schur polynomial theory to show that the $k$-th order minors of any matrix are nonnegative integer linear combinations of row-consecutive $k$-th order minors, meaning minors formed from distinct columns whose consecutive row indices need not coincide across columns. For Hankel matrices, these minors coincide -- up to sign changes arising from column swaps -- with the usual $k$-th order minors with consecutive row indices. Our main result then follows by showing that the sum of certain signed nonnegative integer coefficients equals the corresponding Littlewood--Richardson coefficients. In our problem, the nonnegativity of these coefficients ensures that negatively signed column permutations are cancelled by positively signed ones. Our results also extend naturally to Toeplitz matrices and operators, and we present a partial analogue for circulant matrices.