Algebra Seminar: Khare -- Entrywise functions preserving positivity: from Schur to Schoenberg, to Schur

Apoorva Khare (Indian Institute of Science) 

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

Title: Entrywise functions preserving positivity: from Schur to Schoenberg, to Schur 

Abstract: 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.)