On some Grothendieck expansions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916661578170368 |
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| author | Marberg, Eric Wen, Jiayi |
| author_facet | Marberg, Eric Wen, Jiayi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some Grothendieck expansions Marberg, Eric Wen, Jiayi Combinatorics K-Theory and Homology Representation Theory 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. |
| title | On some Grothendieck expansions |
| topic | Combinatorics K-Theory and Homology Representation Theory |
| url | https://arxiv.org/abs/2412.18963 |