A Survey of the Shi Arrangement

2019 ◽  
pp. 75-113
Author(s):  
Susanna Fishel
Keyword(s):  
Shi Arrangement

scholarly journals The freeness of Ish arrangements

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Daisuke Suyama ◽  
Shuhei Tsujie
Keyword(s):  
Catalan Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Free Arrangement ◽  
International Audience ◽  
New Interpretation ◽  
Necessary And Sufficient ◽  
Shi Arrangement

International audience The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free L’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres $q; t$-Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre.

scholarly journals The Shi arrangement of the type $D_{\ell}$

2012 ◽  
Vol 88 (3) ◽  
pp. 41-45 ◽  
Author(s):  
Ruimei Gao ◽  
Donghe Pei ◽  
Hiroaki Terao
Keyword(s):  
Shi Arrangement

scholarly journals Type C parking functions and a zeta map

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Lattice Paths ◽  
Parking Functions ◽  
Natural Analogue ◽  
International Audience ◽  
Type C ◽  
Shi Arrangement

International audience We introduce type $C$ parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type $C$ parking functions to regions of the Shi arrangement of type $C$, encoded as diagonally labelled ballot paths and endowed with a natural statistic area'. This bijection is a natural analogue of the zeta map of Haglund and Loehr and maps dinv' to area'. We give three different descriptions of it. Nous introduisons les fonctions de stationnement de type $C$, encodées par des chemins étiquetés verticalement et munies d’une statistique dinv'. Nous définissons une bijection entre les fonctions de stationnement de type $C$ et les régions de l’arrangement de Shi de type $C$, encodées par des chemins étiquetés diagonalement et munies d’une statistique naturelle area'. Cette bijection est un analogue naturel à la fonction zeta de Haglund et Loehr, et envoie dinv' sur area'. Nous donnons trois différentes descriptions de celle-ci.

scholarly journals The Shi arrangement and the Ish arrangement

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades
Keyword(s):  
Degrees Of Freedom ◽  
Type A ◽  
Grand Nombre ◽  
Set Partition ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience ◽  
New Interpretation ◽  
Arrangements Of Hyperplanes ◽  
Shi Arrangement

International audience 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. Cet article traite de deux arrangements d'hyperplans. Le premier — arrangement Shi — a été introduit par Jian-Yi Shi pour décrire les cellules de Kazhdan-Lusztig du groupe de Weyl affine de type A. Le deuxième — arrangement Ish — a été récemment défini par le premier auteur pour donner une nouvelle interprétation des nombres q,t-Catalan de Garsia et Haiman. Ici nous définissons une mystérieuse "symétrie combinatoire" entre les deux arrangements et nous montrons que cette symétrie conserve un grand nombre d'informations. Par exemple, les arrangements Shi et Ish ont le même polynôme caractéristique, le même nombre de régions, de régions bornées, de régions dominantes, de régions avec c "plafonds'' et d "degrés de liberté'', etc. En outre, ces résultats se généralisent aux arrangements Shi et Ish "deleted'' correspondant à un sous-graphe arbitraire du graphe complet. Nos preuves reposent sur des étiquetages combinatoires des régions Shi et Ish, et sur une nouvelle statistique associée.

scholarly journals A simple bijection for the regions of the Shi arrangement of hyperplanes

1999 ◽  
Vol 204 (1-3) ◽  
pp. 27-39 ◽  
Author(s):  
Christos A. Athanasiadis ◽  
Svante Linusson
Keyword(s):  
Arrangement Of Hyperplanes ◽  
Shi Arrangement

scholarly journals Bigraphical arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Sam Hopkins ◽  
David Perkinson
Keyword(s):  
American Mathematical Society ◽  
Mathematical Society ◽  
Parking Functions ◽  
Full Version ◽  
Labeled Graphs ◽  
Its Regions ◽  
International Audience ◽  
Related Graph ◽  
Shi Arrangement

International audience We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$

scholarly journals Labeling the Regions of the Type $C_n$ Shi Arrangement

10.37236/3272 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Karola Mészáros
Keyword(s):  
Bijective Proof ◽  
Shi Arrangement

The number of regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ is $(2n+1)^n$. Strikingly, no bijective proof of this fact has been given thus far. The aim of this paper is to provide such a bijection and use it to prove more refined results. We construct a bijection between the regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ and sequences $a_1a_2 \ldots a_n$, where $a_i \in \{-n, -n+1, \ldots, -1, 0, 1, \ldots, n-1, n\}$, $ i \in [n]$. Our bijection naturally restrict to bijections between special regions of the arrangement and sequences with a given number of distinct elements.

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