Séminaire Lotharingien de Combinatoire, 80B.14 (2018), 12 pp.

Apoorva Khare and Terence Tao

Schur Polynomials, Entrywise Positivity Preservers, and Weak Majorization

Abstract. We prove a monotonicity phenomenon for ratios of Schur polynomials. In this we are motivated by - and apply our result to - understanding polynomials and power series that preserve positive semidefiniteness (psd) when applied entrywise to psd matrices. We then extend these results to classify polynomial preservers of total positivity. As a further application, we extend a conjecture of Cuttler, Greene, and Skandera (2011) to obtain a novel characterization of weak majorization using Schur polynomials. Our proofs proceed through a Schur positivity result of Lam, Postnikov, and Pylyavskyy (2007), and computing the leading terms of Schur polynomials.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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