Hybrid Grothendieck polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910967498014720 |
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| author | Guo, Peter L. Kang, Mingyang Liu, Jiaji |
| author_facet | Guo, Peter L. Kang, Mingyang Liu, Jiaji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hybrid Grothendieck polynomials Guo, Peter L. Kang, Mingyang Liu, Jiaji Combinatorics Algebraic Geometry Representation Theory 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. |
| title | Hybrid Grothendieck polynomials |
| topic | Combinatorics Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2505.19072 |