Excited Young diagrams and equivariant Schubert calculus

TL;DR

This paper develops combinatorial and algebraic formulas for the equivariant cohomology of isotropic Grassmannians, introducing excited Young diagrams and factorial Schur functions to compute Schubert classes and singularity multiplicities.

Contribution

It introduces excited Young diagrams and factorial Schur Q- and P-functions as new tools for equivariant Schubert calculus on isotropic Grassmannians.

Findings

Formulas for equivariant Schubert classes using excited Young diagrams.

Giambelli-type formula for equivariant Schubert classes.

Pfaffian formulas for singularity multiplicities.

Abstract

We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'' and the second one is written in terms of factorial Schur $Q$- or $P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.