Root-theoretic Young diagrams and Schubert calculus II

Abstract

ABSTRACT. 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. We then use this translation to prove that the RYD formulas of [SeYo13] for Schubert calculus of the classical (co)adjoint varieties agree with the Pieri rules of [BuKrTa09], which were needed in the proofs of the (co)adjoint formulas.

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

“…In the companion paper [23], it is shown that our rules agree with known Pieri rules [4] (cf. [18,19]).…”

“…The main strategy employed is to prove that the rules of Theorems 1.3 and 1.7 define an associative ring. In the companion paper [Se13+], it is shown that our rules agree with known Pieri rules [BuKrTa09] (cf. [PrRa96,PrRa03]).…”

“…Lemma 5.28. In [Se13+] it is shown that ⋆ agrees with a Pieri rule of [BuKrTa09] for certain generators of H * (LG(2, 2n)) indexed by shapes λ that therefore also generate R. Hence, it follows λ ↔ σ λ defines an isomorphism of the rings R and H ⋆ (LG(2, 2n)).…”

“…, , n − 2, 0|• ↑ we have shown that (R, ⋆) is an associative ring. In [Se13+] it is shown that ⋆ agrees with a Pieri rule of [BuKrTa09] for certain generators of H * (OG(2, 2n)) indexed by shapes λ that therefore also generate R. Hence, it follows λ ↔ σ λ defines an isomorphism of the rings R and H ⋆ (OG(2, 2n)). The case of Theorem 1.7 follows from that of Theorem 5.3 as the computations are the same when λ and µ are both non-Pieri shapes.…”

“…Another piece of evidence for the value of RYDs comes from a new rule for the GL n Belkale-Kumar coefficients [Se13+] (after A. Knutson-K. Purbhoo [KnPu11]). A comparison of RYDs to the indexing system of [BuKrTa09] is also given in loc.…”