Equivariant Pieri Rule for the homology of the affine Grassmannian

TL;DR

This paper presents an explicit equivariant Pieri rule for the homology of the affine Grassmannian, providing new formulas for Schubert class products in types SL_n, Sp_{2n}, and SO_{2n+1}.

Contribution

It introduces a new explicit Pieri rule for affine Grassmannian homology, including a positive formula for SL_n and conjectures for other types, advancing Schubert calculus.

Findings

Explicit Pieri rule for SL_n affine Grassmannian homology

Positivity of the formula for SL_n

Conjectured formulas for Sp_{2n} and SO_{2n+1}

Abstract

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL_n and a similar formula is conjectured for Sp_{2n} and SO_{2n+1}. For SL_n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL_n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.