A Gröbner basis for Kazhdan-Lusztig ideals
A Gröbner basis for Kazhdan-Lusztig ideals
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DOI：
10.1353/ajm.2012.0031
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发表时间：
2009-09
影响因子：
1.7
通讯作者：
Alexander Woo;A. Yong
中科院分区：
文献类型：
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作者：
Alexander Woo;A. Yong
{\it Kazhdan-Lusztig理想}，一个家庭的广义行列式理想研究[吴勇勇'08]，提供了一个明确的选择的坐标和方程编码的一个邻域的一个环面不动点的舒伯特品种上的一个类型$A$旗品种。我们的主要结果是这些理想的Gr\"{o}bner基。这提供了一个单一的几何设置，以透明地解释自然的白日梦的图上的排列，和他们的外观：\开始{itemize} \item组合公式[Fomin-Kirillov '94]舒伯特和Grothendieck多项式的[Lascoux-Sch\"{u}tzenberger '82]; \item等变$K$-理论专业化公式的[Buch-Rim\'{a}nyi '04];和\item的一个积极的组合公式的多重性的舒伯特品种在良好的情况下，包括那些相关的Kazhdan-Lusztig理想是齐次下的标准分级。我们的结果推广了[Knutson-Miller '05]关于Schubert行列式理想的Gr\"{o}bner基定理及其对Schubert多项式单项式正性的几何解释.我们还补充了最近的工作[Knutson '08 $\$ '09]退化的Kazhdan-Lusztig品种一般李型，以及工作[戈尔丁'01]等变本地化和[Lakshmibai-韦曼'90]，[Rosenthal-Zelevinsky '01]，和[Krattenthaler '01]对格拉斯曼多重公式。
{\it Kazhdan-Lusztig ideals}, a family of generalized determinantal ideals investigated in [Woo-Yong~'08], provide an explicit choice of coordinates and equations encoding a neighborhood of a torus-fixed point of a Schubert variety on a type $A$ flag variety. Our main result is a Gr\"{o}bner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the {\it Rothe diagram of a permutation}, and their appearance in: \begin{itemize} \item combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomialsof [Lascoux-Sch\"{u}tzenberger '82]; \item the equivariant $K$-theory specialization formula of [Buch-Rim\'{a}nyi '04]; and \item a positive combinatorial formula for multiplicities of Schubert varieties in good cases, including those for which the associated Kazhdan-Lusztig ideal is homogeneous under the standard grading. \end{itemize} Our results generalize (with alternate proofs) [Knutson-Miller '05]'s Gr\"{o}bner basis theorem for Schubert determinantal ideals and their geometric interpretation of the monomial positivity of Schubert polynomials. We also complement recent work of [Knutson '08 $\&$ '09] on degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian multiplicity formulas.