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VOL. 71 | 2016 Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
Hiraku Abe, Sara Billey

Editor(s) Hiroshi Naruse, Takeshi Ikeda, Mikiya Masuda, Toshiyuki Tanisaki

Abstract

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 Schützenberger 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.

Information

Published: 1 January 2016
First available in Project Euclid: 4 October 2018

zbMATH: 1378.14053
MathSciNet: MR3644818

Digital Object Identifier: 10.2969/aspm/07110001

Subjects:
Primary: 14B05, 14M15, 20F55

Rights: Copyright © 2016 Mathematical Society of Japan

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