A formula for K-theory truncation Schubert calculus

TL;DR

This paper introduces a new formula for truncating Schubert and Grothendieck polynomials, enabling explicit calculations of structure constants in K-theory of flag varieties, generalizing previous cohomology results.

Contribution

It provides subtraction-free formulas for Schubert structure constants in K-theory, extending earlier cohomology formulas and utilizing permutation diagram operations.

Findings

Derived explicit formulas for K-theory Schubert structure constants.

Extended previous cohomology results to K-theory context.

Utilized permutation diagram 'marching' operations for calculations.

Abstract

Define a ``truncation'' $r_{t}(p)$ of a polynomial $p$ in $\{x_1,x_2,x_3,...\}$ as the polynomial with all but the first $t$ variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in $K(Flags({\mathbb C}^n))$, in particular generalizing those from [Kogan '00] in which only cohomology was treated, and from [Buch `02] on the Grassmannian case. The terms of the answer are computed using ``marching'' operations on permutation diagrams.