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The Shi arrangement and the Ish arrangement

Authors: Drew Armstrong and Brendon Rhoades
Journal: Trans. Amer. Math. Soc. 364 (2012), 1509-1528
MSC (2010): Primary 05Exx
Published electronically: October 24, 2011
MathSciNet review: 2869184
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Abstract: This paper is about two arrangements of hyperplanes. The first -- the Shi arrangement -- was introduced by Jian-Yi Shi (1986) to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $ A$. The second -- the Ish arrangement -- was recently defined by the first author, who used the two arrangements together to give a new interpretation of the $ q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious ``combinatorial symmetry'' between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $ c$ ``ceilings'' and $ d$ ``degrees of freedom'', etc. Moreover, all of these results hold in the greater generality of ``deleted'' Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

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Additional Information

Drew Armstrong
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146

Brendon Rhoades
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of Southern California, Los Angeles, California 90089

Received by editor(s): September 8, 2010
Received by editor(s) in revised form: December 7, 2010
Published electronically: October 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.