On some Grothendieck expansions

TL;DR

The paper addresses how to express orthogonal Grothendieck polynomials $\mathcal{G}^{\mathsf{O}}_z$ as nonnegative $\mathcal{G}^{(\beta)}_w$-expansions, with a focus on vexillary and quasi-dominant involutions. It develops new exact formulas (notably for quasi-dominant involutions) and a vexillary expansion theorem (ivex-thm) that yields a monomial-positive description via shiftable subsets, together with shift-invariance results and Grassmannian identities. The approach blends divided difference operators, transition formulas, and combinatorial tools like shiftable left segments and involution pipe dreams to produce stable limits and explicit expansions. These results advance understanding of $K$-theory classes of symmetric matrix Schubert varieties, providing practical methods for computing orthogonal orbit closures and revealing stability phenomena in their $K$-theoretic expansions. The work also clarifies how $GQ_z$ and related symmetric limits arise from vexillary cases, linking to well-known $K$-theoretic Schur $P$- and $Q$-functions.

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 $\mathfrak{G}^{(β)}_w$, 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 $\mathfrak{G}^{(β)}$-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 $\mathfrak{G}^{(β)}$-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for $\mathfrak{G}^{\mathsf{O}}_z$ and its $\mathfrak{G}^{(β)}$-expansion when $z$ is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property.

Figures (5)

• Figure 1: The directed graph $\mathcal{B}_{\mathsf{inv}}^+(z)$ for $z=t_{4}=(1,4)$. The data in each box is $\boxed{w:\operatorname{GC}^{\mathsf{O}}_z(w)}$ with $w$ in inverse one-line notation. The blue vertices are the elements of $\mathcal{B}_{\mathsf{inv}}(z)$. One has $\operatorname{supp}(\operatorname{GC}^{\mathsf{O}}_z) = \mathcal{B}_{\mathsf{inv}}^+(z)$ in this case.
• Figure 2: The directed graph $\mathcal{B}_{\mathsf{inv}}^+(g_3)$, presented as in Figure \ref{['t1n-fig']}.
• Figure 3: The directed graph $\mathcal{B}_{\mathsf{inv}}^+(4321)$, presented as in Figure \ref{['t1n-fig']}.
• Figure 4: Nonzero values of $\operatorname{GC}^{\mathsf{O}}_{w_0}$ for $w_0=n\cdots 321\in I_n$.
• Figure 5: The directed graphs $\mathcal{B}_{\mathsf{inv}}^+(z)$ when $z$ is $g_{23}=(2,4)(3,5)$ (left) and $g_{24}=(2,5)(3,6)(4,7)$ (right), presented using the conventions in Figure \ref{['t1n-fig']}. Here, the grey box indicates the unique element of $\mathcal{B}_{\mathsf{inv}}^+(z)$ not in $\operatorname{supp}(\operatorname{GC}^{\mathsf{O}}_z)$.

Theorems & Definitions (104)

• Example 1.1
• Example 1.2
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['vexweak-prop']}
• Remark 2.5