The Shi arrangement and the Ish arrangement
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چکیده
منابع مشابه
The Shi arrangement and the Ish arrangement
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 prese...
متن کاملShi Threshold Arrangement
Richard Stanley suggested the problem of finding the characteristic polynomial of a certain hyperplane arrangement defined by xi + xj = 0, 1, which is called the Shi threshold arrangement. We present the answer of the problem, using the finite field method.
متن کاملLabeling the Regions of the Type Cn Shi Arrangement
The number of regions of the type Cn Shi arrangement in Rn 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 Cn Shi arrangement in Rn and sequences a1a2 . . . an, where ai ∈ {−n,−n+1, . . . ,−1, 0, 1, . . . , n−1...
متن کاملA Labelling of the Faces in the Shi Arrangement
Let Fn be the face poset of the n-dimensional Shi arrangement, and let Pn be the poset of parking functions of length n with the order defined by (a1, a2, . . . , an) ≤ (b1, b2, . . . , bn) if ai ≤ bi for all i. Pak and Stanley constructed a labelling of the regions in Fn using the elements of Pn. We show that under this labelling, all faces in Fn correspond naturally to closed intervals of Pn,...
متن کاملExtensions of the Shi/ish Duality
We generalize a known bijection between the regions of the Shi and Ish hyperplane arrangements to a bijection between the regions of the nested-Ish and extendedShi arrangements. Although our bijection does not preserve the degrees of freedom statistic, we show that several of the steps do preserve this information, and we give formulas for the number of regions of each arrangement with a given ...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2012
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-2011-05521-2