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

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.