### 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 language | English (US) |
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Title of host publication | FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 51-62 |

Number of pages | 12 |

State | Published - 2011 |

Event | 23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 - Reykjavik, Iceland Duration: Jun 13 2011 → Jun 17 2011 |

### Other

Other | 23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'11 |
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Country | Iceland |

City | Reykjavik |

Period | 6/13/11 → 6/17/11 |

### Fingerprint

### Keywords

- Hyperplane arrangement
- Nonnesting partition
- Product formula

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - The Shi arrangement and the Ish arrangement

AU - Armstrong, Drew

AU - Rhoades, Brendon

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - Hyperplane arrangement

KW - Nonnesting partition

KW - Product formula

UR - http://www.scopus.com/inward/record.url?scp=84860473183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860473183&partnerID=8YFLogxK

M3 - Conference contribution

SP - 51

EP - 62

BT - FPSAC'11 - 23rd International Conference on Formal Power Series and Algebraic Combinatorics

ER -