Abstract
I will report on a recent project [Science 340, 6137] that combines methods from algebraic geometry with a problem that was motivated by theoretical physics, but is at its heart purely tensorial. Indeed, consider a tensor with n indices. There are n ways of associating a matrix with the tensor, by selecting one index and grouping the (n-1) remaining ones together. Each of these matrices comes with a list of singular values. The question now is: What sets of singular values can occur this way and how do they relate to more global properties of the tensor? The problem has applications in quantum mechanics, where every index is associated with a particle and the singular values measure the "degree of uncertainty" ascribed to it. We identify the compatible sets of singular values as a non-abelian moment polytope—an object that is studied in symplectic geometry, asymptotic representation theory, and algebraic geometry. These methods allow us to relate the spectra to the SL-orbit in which the tensor lies. For low-dimensional situations, we can compute the polytopes explicitly and apply them to physical problems.