Department of MathematicsWhich functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, who classified the positivity preservers of matrices of all dimensions. The study of positivity preservers in fixed dimension is harder, and a complete characterization remains elusive to date. In fact it was not known if there exists any analytic preserver with negative coefficients.
We prove such an existence result, and in fact a characterization, for classes of polynomials and power series. Central to the proof are novel determinantal identities involving Schur polynomials, and a monotonicity phenomenon on the positive orthant for ratios of Schur functions (via Lam--Postnikov--Pylyavskyy). If time permits, we will mention an application to a novel characterization of weak majorization, via Schur polynomials and the Harish-Chandra--Itzykson--Zuber integral. This parallels a conjecture of Cuttler--Greene--Skandera. (Joint with Alexander Belton, Dominique Guillot, and Mihai Putinar; and with Terence Tao.)