Equivariant Pieri rules for isotropic Grassmannians

Abstract

Abstract. We give a Pieri rule for the torus-equivariant cohomology of (submaximal) Grassmannians of Lie types B, C, and D. To the authors' best knowledge, our rule is the first manifestly positive formula, beyond the equivariant Chevalley formula. We also give a simple proof of the equivariant Pieri rule for the ordinary (type A) Grassmannian.

The A.B.C.Ds of Schubert calculus

“…Here is another consequence of Theorem 1.1. C. Li-V. Ravikumar [30] prove equivariant Pieri rules for (submaximal) isotropic Grassmannians of classical type B, C, D. Their type B and C rules are proved by separate geometric analyses. Theorem 1.1 immediately implies a Pieri rule for type C from the type B rule (or vice versa).…”

A geometric proof of an equivariant Pieri rule for flag manifolds

“…We prove Theorem 1.2 using the method of [8] and exploiting the explicit description of certain Richardson varieties and their projections in [11].…”

“…Here, ν is the indexing set defined in the statement of Theorem 1.2. As in [8], Z u w equals the projected Richardson variety coming from the single point Richardson variety e k | k ∈ ν in the Grassmannian G(m+p−q, n), via the standard maps (4) from Fℓ(1, m+p−q; n). By [8,Proposition 4.2], c u w,r(m,p) equals N ν ν,q , the localization of the special Schubert class of codimension q in the Grassmannian G(m+p−q, n) at the torus-fixed point e k | k ∈ ν .…”

“…As in [8], Z u w equals the projected Richardson variety coming from the single point Richardson variety e k | k ∈ ν in the Grassmannian G(m+p−q, n), via the standard maps (4) from Fℓ(1, m+p−q; n). By [8,Proposition 4.2], c u w,r(m,p) equals N ν ν,q , the localization of the special Schubert class of codimension q in the Grassmannian G(m+p−q, n) at the torus-fixed point e k | k ∈ ν . The indexing set ν is a Schubert symbol in [8], and the Grassmannian permutation associated to ν is obtained by sortingν := {w (1) From the proof of Lemma 15 in [11], there are partitions µ ⊂ λ for G(m, n), and a permutation ω ∈ S n such that c u w,r(m,p) = ω(c λ µ,(p) ).…”

On equivariant quantum Schubert calculus for G/P

“…It is obtained by simplifying Robinson's Pieri rule in a purely combinatorial way. Nevertheless, our formulation has inspired the second author and Ravikumar to find an equivariant Pieri rule for Grassmannians of all classical Lie types [38] in a geometric way. Definition 3.14.…”