Secants of minuscule and cominuscule minimal orbits

We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian

Algebraic Methods of Classifying Directed Graphical Models

Directed acyclic graphical models (DAGs) are often used to describe common structural properties in a family of probability distributions. This paper addresses the question of classifying DAGs up to an isomorphism. By considering Gaussian densities, the question reduces to verifying equality of certain algebraic varieties. A question of computing equations for these varieties has been previously raised in the literature. Here it is shown that the most natural method adds spurious components with singular principal minors, proving a conjecture of Sullivant. This characterization is used to establish an algebraic criterion for isomorphism, and to provide a randomized algorithm for checking that criterion. Results are applied to produce a list of the isomorphism classes of tree models on 4,5, and 6 nodes. Finally, some evidence is provided to show that projectivized DAG varieties contain useful information in the sense that their relative embedding is closely related to efficient inference