Double theta polynomials and equivariant Giambelli formulas

TL;DR

This paper introduces double theta polynomials using Young's raising operators, providing new Giambelli formulas for equivariant Schubert classes in symplectic Grassmannians and a novel presentation of their cohomology ring.

Contribution

It develops a unified framework for double theta polynomials that generalize existing polynomials and derives explicit Giambelli formulas and ring presentations for symplectic Grassmannians.

Findings

Double theta polynomials unify several polynomial families.

New Giambelli formulas for equivariant Schubert classes.

A new presentation of the cohomology ring of symplectic Grassmannians.

Abstract

We use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.