On some Grothendieck expansions

Abstract

The orthogonal and symplectic groups act on the complete flag variety with finitely many orbits. The orthogonal Grothendieck polynomials ${\mathfrak{G}}_z^{\mathsf{O}}$ and symplectic Grothendieck polynomials ${\mathfrak{G}}_z^{\mathsf{Sp}}$ are distinguished representatives for the $K$-theory classes of the corresponding orbit closures. There is a simple formula to expand ${\mathfrak{G}}_z^{\mathsf{Sp}}$ as a linear combination of Grothendieck polynomials ${\mathfrak{G}}_w^{({\hat{I}}^2)}$, which represent the $K$-theory classes of Schubert varieties. Although the constructions of ${\mathfrak{G}}_z^{\mathsf{Sp}}$ and ${\mathfrak{G}}_z^{\mathsf{O}}$ are similar, finding the ${\mathfrak{G}}^{({\hat{I}}^2)}$-expansion of ${\mathfrak{G}}_z^{\mathsf{O}}$ or even computing ${\mathfrak{G}}_z^{\mathsf{O}}$ is much harder. If $z$ is vexillary then ${\mathfrak{G}}_z^{\mathsf{O}}$ has a nonnegative ${\mathfrak{G}}^{({\hat{I}}^2)}$-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for ${\mathfrak{G}}_z^{\mathsf{O}}$ and its ${\mathfrak{G}}^{({\hat{I}}^2)}$-expansion when $z$ is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property.

51 pages, 5 figures; v2: minor improvements and corrections; v3: some corrections, added exposition, updated references, final version