Twisted factorial Grothendieck polynomials and equivariant $K$-theory of weighted Grassmann orbifolds

TL;DR

This paper introduces twisted factorial Grothendieck polynomials to explicitly describe Schubert classes and structure constants in the equivariant K-theory of weighted Grassmann orbifolds.

Contribution

It defines new symmetric polynomials that represent Schubert classes and provides explicit formulas for restrictions and structure constants in this setting.

Findings

Twisted factorial Grothendieck polynomials represent Schubert classes.

Explicit formulas for restrictions to torus fixed points.

Structure constants in K-theory are explicitly described.

Abstract

In this paper, we provide an explicit description of the Schubert classes in the equivariant $K$-theory of weighted Grassmann orbifolds. We introduce the `twisted factorial Grothendieck polynomials', a family of symmetric polynomials by specializing the factorial Grothendieck polynomials, and prove that they represent the Schubert classes in the equivariant $K$-theory of the weighted Grassmann orbifolds. We give an explicit formula for the restriction of the Schubert classes to any torus fixed point in terms of twisted factorial Grothendieck polynomials. We give an explicit formula for the structure constants with respect to the Schubert basis in the equivariant $K$-theory of weighted Grassmann orbifolds. Eminently, we describe `twisted Grothendieck polynomials' and prove that these represent the Schubert classes in the $K$-theory of the weighted Grassmann orbifold. As a consequence, we describe the structure constants in the $K$-theory of weighted Grassmann orbifolds.