Spectrahedral representability and Matroids

For each hyperbolic polynomial h, there is an associated closed convex cone called the hyperbolicity cone of h, whose interior contains all the directions e for which h is hyperbolic. Moreover, a convex cone is called spectrahedral, if it can be described by linear matrix inequalities with symmetric matrices. Is every hyperbolicity cone spectrahedral? This is the question generalized Lax conjecture considers and posits.

Choe et. al. in 2004 showed that the support of each homogeneous multiaffine polynomial with the half-plane property (such a polynomial is hyperbolic) is the collection of bases of some matroid M. Their result lets us switch to the combinatorial world, search for matroids corresponding to a hyperbolic polynomial, and consider the spectrahedral representability in that setting.

In this talk, we take this matroid theoretic approach, and present our results on the spectrahedral representability being closed under taking minors. We continue with the classification of matroids on 8 elements with respect to the half-plane property.