Bumpless pipe dreams and alternating sign matrices
Bumpless pipe dreams and alternating sign matrices
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无扰动白日梦和交替符号矩阵
DOI：
10.1016/j.jcta.2021.105470
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Anna Weigandt
中科院分区：
文献类型：
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作者：
Anna Weigandt
在关于无限旗簇的研究中，Lam、Lee 和 Shimozono（2018）引入了称为无扰动白日梦的对象，并用它们给出了双舒伯特多项式的公式。我们将此公式扩展到 K 理论的设置，给出了双格罗滕迪克多项式的表达式作为更大类无扰动白日梦的总和。我们的证明依赖于 Lascoux（2002）未发表的手稿中发现的技术。拉斯科展示了如何将双格罗腾迪克多项式写为交替符号矩阵的和。我们解释如何将 Lam-Lee-Shimozono 公式视为拉斯科交替符号矩阵公式的变相特例。 Knutson、Miller 和 Yong (2009) 给出了向量 Grothendieck 多项式的表格公式。我们通过显示向量标记的无扰动白日梦和标记的集值画面处于保重双射来恢复这个公式。最后，我们在赫克无波澜的白日梦和减少的画面之间给出了双射。对 Edelman-Greene 无扰动白日梦的双射限制解决了 Lam、Lee 和 Shimozono 的问题。
In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018) introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of Lascoux (2002). Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula. Knutson, Miller, and Yong (2009) gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono.