Principal minors and rhombus tilings

Abstract

The algebraic relations between the principal minors of an $n\times n$ matrix are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by adding in certain \emph{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 $n\times n$ 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 $\mathbb R, \mathbb C$ or $\mathbb H$ the quaternions. Moreover by further specialization we can parametrize the space of \emph{positive definite} matrices over these rings.