Combinatorics Seminar: Preservers of positivity and total positivity in fixed dimension

Seminar | December 7 | 2-3 p.m. | 939 Evans Hall

 Apoorva Khare, Indian Institute of Science

 Department of Mathematics

We discuss which functions preserve the notions of positive semidefiniteness and total positivity, when applied entrywise to matrices in a fixed dimension. 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 analytic 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, and a Schur positivity result of Lam-Postnikov-Pylyavskyy. An application is a novel characterization of weak majorization via totally positive matrices; this extends a conjecture of Cuttler-Greene-Skandera. We then completely classify the preservers of total positivity (TP) in each fixed dimension. In particular, for sizes 4-by-4 or larger, every such preserver must be constant or linear. (Joint with Alexander Belton, Dominique Guillot, and Mihai Putinar; and with Terence Tao.)