Hybrid Grothendieck polynomials

TL;DR

The paper introduces hybrid Grothendieck polynomials ${G}_{\lambda/\mu}(\mathbf{x};\mathbf{t};\mathbf{w})$ as weight-generating functions over set-valued reverse plane partitions, unifying stable and dual stable Grothendieck polynomials. It establishes symmetry in $\mathbf{x}$, constructs a GL$_n$-crystal on SVRPPs to obtain a Schur-expansion, and proves SNP for straight shapes via a crystal-theoretic framework and Nguyen’s criterion. Using Fomin–Greene’s noncommutative Schur theory, it derives a combinatorial formula for the omega-image ${J}_{\lambda/\mu}(\mathbf{x};\alpha;\beta)$ in terms of marked multiset-valued tableaux, thereby unifying structures from weak and valued-set tableaux. The work also discusses several open problems and conjectures, including basis, determinantal formulas, alternate bases, integrable lattice models, and deeper coincidence phenomena among hybrids, illustrating the broad reach and potential extensions of the theory.

Abstract

For a skew shape $λ/μ$, we define the hybrid Grothendieck polynomial $${G}_{λ/μ}(\textbf{x};\textbf{t};\textbf{w}) =\sum_{T\in \mathrm{SVRPP}(λ/μ)} \textbf{x}^{\mathrm{ircont}(T)}\textbf{t}^{\mathrm{ceq} (T)}\textbf{w}^{\mathrm{ex}(T)}$$ as a weight generating function over set-valued reverse plane partitions of shape $λ/μ$. It specializes to \begin{itemize} \item[(1)] the refined stable Grothendieck polynomial introduced by Chan--Pflueger by setting all $t_i=0$; \item[(2)] the refined dual stable Grothendieck polynomial introduced by Galashin--Grinberg--Liu by setting all $w_i=0$. \end{itemize} We show that ${G}_{λ/μ}(\textbf{x};\textbf{t};\textbf{w})$ is symmetric in the $\textbf{x}$ variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of ${G}_{λ/μ}(\textbf{x};\textbf{t};\textbf{w})$ in the basis of Schur functions, extending previous work by Monical--Pechenik--Scrimshaw and Galashin. Based on the Schur expansion, we deduce that hybrid Grothendieck polynomials of straight shapes have saturated Newton polytopes. Finally, using Fomin--Greene's theory on noncommutative Schur functions, we give a combinatorial formula for the image of ${G}_{λ/μ}(\textbf{x};\textbf{t};\textbf{w})$ (in the case $t_i=α$ and $w_i=β$) under the omega involution on symmetric functions. The formula unifies the structures of weak set-valued tableaux and valued-set tableaux introduced by Lam--Pylyavskyy. Several problems and conjectures are motivated and discussed.

Figures (19)

• Figure 1: $T_1$, $T_2$ and $T_3$ are SVRPP's. Furthermore, $T_2$ is a set-valued tableau, and $T_3$ is a reverse plane partition. Here we have omitted the braces $\{\ \}$ for sets as well as the commas between numbers.
• Figure 2: Specializations of hybrid Grothendieck polynomials.
• Figure 3: The Newton polytope of $G_{(4,2,1)}(\mathbf{x}_2;\mathbf{1};\mathbf{1})$.
• Figure 4: The descent-resolution operations.
• Figure 5: Examples of the descent-resolution operations.

Theorems & Definitions (53)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5