Hybrid Grothendieck polynomials

TL;DR

This paper introduces hybrid Grothendieck polynomials as a new family unifying previous polynomials, proves their symmetry and Schur expansion, and explores their geometric and combinatorial properties.

Contribution

It defines hybrid Grothendieck polynomials, establishes their symmetry, provides a Schur expansion via crystal structures, and offers a combinatorial formula linking to noncommutative Schur functions.

Findings

Proved symmetry in x variables.

Derived Schur function expansion.

Established saturated Newton polytopes for straight shapes.

Abstract

For a skew shape $\lambda/\mu$, we define the hybrid Grothendieck polynomial $${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w}) =\sum_{T\in \mathrm{SVRPP}(\lambda/\mu)} \textbf{x}^{\mathrm{ircont}(T)}\textbf{t}^{\mathrm{ceq} (T)}\textbf{w}^{\mathrm{ex}(T)}$$ as a weight generating function over set-valued reverse plane partitions of shape $\lambda/\mu$. 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 ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ is symmetric in the $\textbf{x}$ variables. By building a crystal structure on set-valued reverse plane partitions, we obtain the expansion of ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ 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 ${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w})$ (in the case $t_i=\alpha$ and $w_i=\beta$) 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.