The Shi arrangement and the Ish arrangement

Drew Armstrong, Brendon Rhoades

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationFPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics
Pages51-62
Number of pages12
StatePublished - 2011
Event23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 - Reykjavik, Iceland
Duration: Jun 13 2011Jun 17 2011

Other

Other23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11
CountryIceland
CityReykjavik
Period6/13/116/17/11

Fingerprint

Arrangement
Arrangement of Hyperplanes
Affine Weyl Groups
Symmetry
Set Partition
Catalan number
Ceiling
Characteristic polynomial
Complete Graph
Statistic
Subgraph
Degree of freedom
Cell
Arbitrary

Keywords

  • Hyperplane arrangement
  • Nonnesting partition
  • Product formula

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Armstrong, D., & Rhoades, B. (2011). The Shi arrangement and the Ish arrangement. In FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics (pp. 51-62)

The Shi arrangement and the Ish arrangement. / Armstrong, Drew; Rhoades, Brendon.

FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics. 2011. p. 51-62.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Armstrong, D & Rhoades, B 2011, The Shi arrangement and the Ish arrangement. in FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics. pp. 51-62, 23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11, Reykjavik, Iceland, 6/13/11.
Armstrong D, Rhoades B. The Shi arrangement and the Ish arrangement. In FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics. 2011. p. 51-62
Armstrong, Drew ; Rhoades, Brendon. / The Shi arrangement and the Ish arrangement. FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics. 2011. pp. 51-62
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