Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
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Lakshmibai-Sandhya 定理的后果：舒伯特微积分和相关几何中排列模式的普遍性
DOI：
10.2969/aspm/07110001
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Sara C. Billey
中科院分区：
文献类型：
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作者：
Hiraku Abe;Sara C. Billey
1990 年，Lakshmibai 和 Sandhya 发表了使用模式避免概念的旗形流形中奇异舒伯特簇的表征。这是第一次使用模式避免来表征舒伯特簇的几何特性。他们的结果与 Haiman、Ryan 和 Wolper 的工作密切相关，但 Lakshmibai-Sandhya 是第一个准确使用该语言的人。 Knuth、Pratt、Tarjan 和其他人在 20 世纪 60 年代和 1970 年代曾使用排列中的模式避免来描述计算机科学中的排序算法。 Lascoux 和 Sch$\text{\"u}$tzenberger 在 20 世纪 80 年代也使用模式避免来表征向量排列。现在，舒伯特簇的许多几何性质都使用模式避免作为表征方法，包括 Gorenstein、阶乘、局部完全交集和 Kazhdan-Lusztig 多项式的性质。这些就是我们所说的 Lakshmibai-Sandhya 定理的推论。调查了该领域的许多漂亮的结果、概括和遗留问题。我们强调了在线性时间算法方面使用模式避免特征的优点，以及通过 Tenner 排列模式避免数据库轻松访问文献的方法。这项调查基于第二作者在 2012 年日本数学会暑期学校以舒伯特微积分为主题的讲座。
In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was used to characterize geometrical properties of Schubert varieties. Their results are very closely related to work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya were the first to use that language exactly. Pattern avoidance in permutations was used historically by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize sorting algorithms in computer science. Lascoux and Sch$\text{\"u}$tzenberger also used pattern avoidance to characterize vexillary permutations in the 1980's. Now, there are many geometrical properties of Schubert varieties that use pattern avoidance as a method for characterization including Gorenstein, factorial, local complete intersections, and properties of Kazhdan-Lusztig polynomials. These are what we call consequences of the Lakshmibai-Sandhya theorem. We survey the many beautiful results, generalizations, and remaining open problems in this area. We highlight the advantages of using pattern avoidance characterizations in terms of linear time algorithms and the ease of access to the literature via Tenner's Database of Permutation Pattern Avoidance. This survey is based on lectures by the second author at Osaka, Japan 2012 for the Summer School of the Mathematical Society of Japan based on the topic of Schubert calculus.