Non-representable hyperbolic matroids

We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vámos matroid and the generalized Vámos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre-Turán inequality and an inequality due to Jensen. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.