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

Abstract

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.

Root-theoretic Young diagrams and Schubert calculus II

“…Overview. In [SeYo13], A. Yong and the author study root-theoretic Young diagrams (RYDs), which are one of several natural choices of indexing set for the Schubert subvarieties of generalized flag varieties. The thesis of that paper and the present one is that RYDs are useful for studying general patterns in Schubert combinatorics in a uniform manner.…”

“…Kresch-H. Tamvakis [BuKrTa09] define an indexing set for the Schubert varieties of non-maximal isotropic Grassmannians, and use this indexing set to give particularly nice Pieri rules for the Schubert calculus of these spaces. The Schubert calculus formulas of [SeYo13] for the (co)adjoint varieties of classical Lie type were discovered using the RYD model to index Schubert varieties. The proof of these formulas we requires Pieri rules for these (co)adjoint varieties, the most interesting of which belong to the family of non-maximal isotropic Grassmannians.…”

“…, k d−1 . In the case of F k , the RYDs of [SeYo13] are the inversion sets of the elements of S k n in the poset Ω GLn of positive roots of GL n . Let Y k denote the set of RYDs for F k .…”

“…The elements of these sets are certain signed permutations corresponding to Weyl group cosets, and are described explicitly in Section 3. For OG(k, 2n+1)/LG(k, 2n) (respectively, OG(k, 2n)), the RYDs of [SeYo13] are the inversion sets of the elements of W OG(k,2n+1) (respectively, W OG(k,2n) ) in the type B root poset Ω SO 2n+1 (respectively, type D root poset Ω SO 2n ). Denote the set of RYDs associated to W OG(k,2n+1) (respectively, W OG(k,2n) ) by Y OG(k,2n+1) (respectively, Y OG(k,2n) ).…”

“…

We continue the study of root-theoretic Young diagrams (RYDs) from [SeYo13]. We provide an RYD formula for the GL n Belkale-Kumar product, after [KnPu11], and we give a translation of the indexing set of [BuKrTa09] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs.

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