Principal minors and rhombus tilings

The algebraic relations between the principal minors of an n× n matrix are somewhat mysterious, see e.g. [LS09]. We show, however, that by adding in certain almost principal minors, the relations are generated by a single relation, the so-called hexahedron relation, which is a composition of six cluster mutations. We give in particular a Laurent-polynomial parameterization of the space of n× n matrices, whose parameters consist of certain principal and almost principal minors. The parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex 2n-gon. A matrix is associated to an equivalence class of tilings, all related to each other by Yang-Baxter-like transformations. By specializing the initial data we can similarly parametrize the space of Hermitian symmetric matrices over R,C or H the quaternions. Moreover by further specialization we can parametrize the space of positive definite matrices over these rings.