Determinantal representations of adjoints of polytopes

The adjoint polynomial of a polytope is the numerator of its canonical form. A mathematically (and, it seems, physically) interesting question is whether adjoints have determinantal representations, that is, can be written as determinants of matrices of linear forms. In this talk I will report on this question. In particular, I will discuss when this is true (and when this is false), present constructive methods to obtain determinantal representations of adjoints and some counterexamples. Our attention will be mostly focused on two- and three-dimensional polytopes as well as on ABHY associahedra. This is joint work in progress with Clemens Brüser and Mario Kummer.