Skip to Main content Skip to Navigation
Conference papers

The Shi arrangement and the Ish arrangement

Abstract : This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi 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 labellings of Shi and Ish regions and a new set partition-valued statistic on these regions.
Complete list of metadatas

Cited literature [13 references]  Display  Hide  Download

https://hal.inria.fr/hal-01215091
Contributor : Coordination Episciences Iam <>
Submitted on : Tuesday, October 13, 2015 - 3:06:36 PM
Last modification on : Wednesday, June 26, 2019 - 2:48:03 PM
Long-term archiving on: : Thursday, April 27, 2017 - 12:14:47 AM

File

dmAO0106.pdf
Publisher files allowed on an open archive

Identifiers

  • HAL Id : hal-01215091, version 1

Collections

Citation

Drew Armstrong, Brendon Rhoades. The Shi arrangement and the Ish arrangement. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.51-62. ⟨hal-01215091⟩

Share

Metrics

Record views

160

Files downloads

892