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A Survey of the Shi Arrangement

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Recent Trends in Algebraic Combinatorics

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 16))

Abstract

In [58], Shi proved Lusztig’s conjecture that the number of two-sided cells for the affine Weyl group of type \(A_{n-1}\) is the number of partitions of n. As a byproduct, he introduced the Shi arrangement of hyperplanes and found a few of its remarkable properties. The Shi arrangement has since become a central object in algebraic combinatorics. This article is intended to be a fairly gentle introduction to the Shi arrangement, intended for readers with a modest background in combinatorics, algebra, and Euclidean geometry.

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Acknowledgements

The author would like to thank Hélène Barcelo, Gizem Karaali, and Rosa Orellana for allowing her to write an article for this AWM series. She would like to thank Matthew Fayers, Sarah Mason, and Jian-Yi Shi for comments on the manuscript, and Nathan Williams and Brendon Rhoades for suggesting several related papers. She would like to thank Patrick Headley for help with his thesis. The anonymous referees’ comments helped enormously to improve exposition. This work was supported by a grant from the Simons Foundation (#359602, Susanna Fishel).

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  1. School of Mathematical and Statistical Sciences, Arizona State University, P. O. Box 871804, Tempe, AZ, 85287-1804, USA

    Susanna Fishel

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  1. Susanna Fishel
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Correspondence to Susanna Fishel .

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  1. Mathematical Sciences Research Institute , Berkeley, CA, USA

    Hélène Barcelo

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Gizem Karaali

  3. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Rosa Orellana

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Fishel, S. (2019). A Survey of the Shi Arrangement. In: Barcelo, H., Karaali, G., Orellana, R. (eds) Recent Trends in Algebraic Combinatorics. Association for Women in Mathematics Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-05141-9_3

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