Which 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, and has also recently received renewed attention due to its applications in high-dimensional statistics. However, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. We provide the first such characterization, for classes of polynomials. Remarkably, the proof involves representation-theoretic techniques, via Schur polynomials.

Our result can be reformulated as a variational problem involving Rayleigh quotients, and leads to a host of interesting consequences, including thresholds for coefficients of arbitrary analytic functions preserving positivity, and linear matrix inequalities for Hadamard powers. We also provide a hitherto unexplored "Schubert-cell type" stratification of the cone of psd matrices, obtained from a novel block decomposition of an arbitrary psd matrix. (Joint with A. Belton, D. Guillot, and M. Putinar.)