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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) 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.