Root-theoretic Young diagrams and Schubert calculus: Planarity and the adjoint varieties

We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H. Thomas and the second author using work of R. Proctor). Our results build on earlier Pieri-type rules of P. Pragacz-J. Ratajski and of A. Buch-A. Kresch-H. Tamvakis. Specifically, our formula for OG(2, 2 n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess both uniform and non-uniform features. Using these classical type rules, as well as results of P.-E. Chaput-N. Perrin in the exceptional types, we suggest a connection between polytopality of the set of nonzero Schubert structure constants and planarity of the diagrams. This is an addition to work of A. Klyachko and A. Knutson-T. Tao on the Grassmannian and of K. Purbhoo-F. Sottile on cominuscule varieties, where the diagrams are always planar.