On some Grothendieck expansions
| Autor: | Marberg, Eric, Wen, Jiayi |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathfrak{G}^{(\beta)}_w$, 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 $\mathfrak{G}^{(\beta)}$-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 $\mathfrak{G}^{(\beta)}$-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for $\mathfrak{G}^{\mathsf{O}}_z$ and its $\mathfrak{G}^{(\beta)}$-expansion when $z$ is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property. Comment: 47 pages, 2 figures |
| Databáze: | arXiv |
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