Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities

A (special case of a) fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer n > 1, if the entrywise application of a smooth function f : (0,∞) → R preserves the set of n × n positive semidefinite matrices with positive entries, then the first n derivatives of f are non-negative on (0,∞). In a recent joint work with Belton– Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and further used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In parallel, in recent works with Belton–Guillot–Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, ‘real analytic versions’ at the origin of the Horn–Loewner condition, and using this discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a ‘Master Theorem’ (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n nonzero Taylor coefficients at individual points rather than on all of (0,∞). A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials naturally arise from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series 1/(1− ujvk)), as well as of a determinant of Frobenius [J. reine angew. Math. 1882] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.