Séminaire Lotharingien de Combinatoire, 80B.14 (2018), 12 pp.
Apoorva Khare and Terence Tao
Schur Polynomials, Entrywise Positivity Preservers, and Weak
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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