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(2,2n)OG(2,2n) 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.