  A classical result by Schoenberg (1942) identifies all real-valued functions
that preserve positive semidefiniteness (psd) when applied entrywise to
matrices of arbitrary dimension. Schoenberg's work has continued to attract
significant interest, including renewed recent attention due to applications in
high-dimensional statistics. However, despite a great deal of effort in the
area, an effective characterization of entrywise functions preserving
positivity in a fixed dimension remains elusive to date. As a first step, we
characterize new classes of polynomials preserving positivity in fixed
dimension. The proof of our main result is representation theoretic, and
employs Schur polynomials. An alternate, variational approach also leads to
several interesting consequences including (a) a hitherto unexplored Schubert
cell-type stratification of the cone of psd matrices, (b) new connections
between generalized Rayleigh quotients of Hadamard powers and Schur
polynomials, and (c) a description of the joint kernels of Hadamard powers.
