On some Grothendieck expansions

TL;DR

This paper develops new formulas for orthogonal Grothendieck polynomials and their expansions in $K$-theory, especially for vexillary cases, revealing stability properties and advancing understanding of orbit closure classes.

Contribution

It introduces explicit formulas and stability results for the $eta$-expansion of orthogonal Grothendieck polynomials in vexillary cases.

Findings

Derived new formulas for $rak{G}^{ ext{O}}_z$

Proved nontrivial stability of the $eta$-expansion

Extended understanding of orbit closure classes in $K$-theory

Abstract

The complete flag variety admits a natural action by both the orthogonal group and the symplectic group. Wyser and Yong defined orthogonal Grothendieck polynomials $\mathfrak{G}^{\mathsf{O}}_z$ and symplectic Grothendieck polynomials $\mathfrak{G}^{\mathsf{Sp}}_z$ as the $K$-theory classes of the corresponding orbit closures. There is an explicit formula to expand $\mathfrak{G}^{\mathsf{Sp}}_z$ as a nonnegative sum of Grothendieck polynomials $\mathfrak{G}^{(\beta)}_w$, which represent the $K$-theory classes of Schubert varieties. Although the constructions of $\mathfrak{G}^{\mathsf{Sp}}_z$ and $\mathfrak{G}^{\mathsf{O}}_z$ are similar, finding the $\mathfrak{G}^{(\beta)}$-expansion of $\mathfrak{G}^{\mathsf{O}}_z$ or even computing $\mathfrak{G}^{\mathsf{O}}_z$ is much harder. If $z$ is vexillary then $\mathfrak{G}^{\mathsf{O}}_z$ has a nonnegative $\mathfrak{G}^{(\beta)}$-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for $\mathfrak{G}^{\mathsf{O}}_z$ and its $\mathfrak{G}^{(\beta)}$-expansion when $z$ is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property.