Cauchy Formulas and Billey's Formulas for Generalized Grothendieck polynomials

TL;DR

This paper explores generalized Grothendieck polynomials across all types, establishing Cauchy formulas, a K-theoretic comodule structure, and a combinatorial localization formula extending Billey's work.

Contribution

It introduces new Cauchy formulas and a combinatorial localization formula for generalized Grothendieck polynomials, expanding their theoretical framework.

Findings

Derived K-theoretic comodule structure map for flag varieties.

Established Cauchy formulas for all types of generalized Grothendieck polynomials.

Provided a combinatorial formula generalizing Billey's formula for Schubert class localization.

Abstract

We study the generalized double $\beta$-Grothendieck polynomials for all types. We study the Cauchy formulas for them. Using this, we deduce the K-theoretic version of the comodule structure map $\alpha^*: K(G/B)\to K(G)\otimes K(G/B)$ induced by the group action map for reductive group $G$ and its flag variety $G/B$. Furthermore, we give a combinatorial formula to compute the localization of Schubert classes as a generalization of Billey's formula.