A Pieri-type formula for isotropic flag manifolds

Abstract

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal{B}}_{\infty }$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show these identities follow from the Pieri-type formula, except some `hidden symmetries' of the structure constants. Our analysis leads to a new partial order on the Coxeter group ${\mathcal{B}}_{\infty }$ and formulas for many of these structure constants.

References in corpus (3)

• Schubert polynomials, the Bruhat order, and the geometry of flag manifolds
• A Monoid for the Universal K-Bruhat Order
• Pieri-type formulas for maximal isotropic Grassmannians via triple intersections

Cited by in corpus (12)

• Quantum Pieri rules for isotropic Grassmannians
• Non-commutative Pieri operators on posets
• Degeneracy Loci Classes in $K$-theory - Determinantal and Pfaffian Formula -
• Gröbner geometry of Schubert polynomials
• A Monoid for the Universal K-Bruhat Order
• Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials
• Littlewood-Richardson coefficients for reflection groups
• Quantum Pieri rules for tautological subbundles
• The A.B.C.Ds of Schubert calculus
• A quantum Murnaghan--Nakayama rule for the flag manifold
• Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties
• The isomorphism problem for Schubert varieties