University of Sydney Algebra Seminar

Apoorva Khare (Indian Institute of Science)

Friday 6 April, 12-1pm, Place: Carslaw 375

Entrywise polynomials preserving positivity: from Schur to Schoenberg, to Schur

We discuss which functions preserve positive semidefiniteness, when applied entrywise to matrices in both the dimension-free and fixed dimension settings. This question has a long history, starting from 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 polynomial preserver with negative coefficients. We prove such an existence result, and in fact a characterization, for classes of polynomials. Central to the proof are novel determinantal identities involving Schur polynomials. We then mention an application to a novel characterization of weak majorization, via totally positive matrices and the Harish-Chandra--Itzykson--Zuber integral. This extends a conjecture of Cuttler-Greene-Skandera. (Joint with Alexander Belton, Dominique Guillot, and Mihai Putinar; and with Terence Tao.)