Secant dimensions of minimal orbits: computations and conjectures

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits we give a short proof of the relation—known from the work of Ehrenborg, Catalisano–Geramita–Gimigliano, and Sturmfels–Sullivant—between the existence of certain codes and nondefectiveness of certain higher secant varieties.