Schur polynomials and matrix positivity preservers

TL;DR

This paper characterizes new classes of polynomials that preserve positive semidefiniteness in fixed dimensions, using Schur polynomials and representation theory, with implications for matrix analysis and high-dimensional statistics.

Contribution

It introduces a novel characterization of positivity-preserving polynomials in fixed dimension employing Schur polynomials and representation theoretic methods.

Findings

New classes of positivity-preserving polynomials characterized

Schubert cell-type stratification of psd cone discovered

Connections between Rayleigh quotients, Hadamard powers, and Schur polynomials established

Abstract

A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.