Polyhedral models for K-theory of toric and flag varieties

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov–Khovanskii type presentation to the case of Ktheory of toric and flag varieties. First we study the Gorenstein duality quotients of the group algebra of free abelian group (possibly of infinite rank). Then we specialize to the K-ring of integer (virtual) polytopes with a fixed normal fan. Finally we show that the K-theory of toric and flag varieties can be realized as polytope K-rings and describe the classes of toric orbits or Schubert varieties in them.