Results 11 to 20 of about 904,790 (255)
The Shi arrangement and the Ish arrangement [PDF]
Discrete Mathematics & Theoretical Computer Science, 2011 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.
Drew Armstrong, Brendon Rhoades
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A simple bijection for the regions of the Shi arrangement of hyperplanes [PDF]
Discrete Mathematics, 1999 The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of $
Svante Linusson, Christos Athanasiadis
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On the Falk Invariant of Shi and Linial Arrangements [PDF]
Discrete and Computational Geometry, 2021 It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete
Michele Torielli, Weili Guo
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Freeness for restriction arrangements of the extended Shi and Catalan arrangements
Discrete Mathematics, 2023 The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension in which the cone of the extended Shi arrangement of type A is hereditarily free.
Shuhei Tsujie, Norihiro Nakashima
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Hyperfactord of Shi arrangement Sh(A2) and Sh(A3)
Al-Mustansiriyah Journal of Science, 2022 In this paper, we introduce the region and the faces poset of shi arrangement that J. Y. Shi firstly introduced it. This is an affine arrangement, each of whose hyperplane is parallel to some"hyperplane of Coxeter arrangement"(Braid arrangement), the ...
Alaa A. A. Al-Mujmaey +1 more
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Bijections of dominant regions in the $m$-Shi arrangements of type $A$, $B$ and $C$ [PDF]
Discrete Mathematics & Theoretical Computer Science, 2015 In the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(
Myrto Kallipoliti, Eleni Tzanaki
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The Shi arrangement of the type $D_{\ell}$
Proceedings of the Japan Academy Series A: Mathematical Sciences, 2012 In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type $D_\ell$ explicitly.
Ruimei Gao, Hiroaki Terao
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Ad-nilpotent ideals and the Shi arrangement
Journal of Combinatorial Theory - Series A, 2013 We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical ...
Chao-Ping Dong
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Hyperplane arrangements between Shi and Ish [PDF]
Electronic Notes in Discrete Mathematics, 2018 Abstract We introduce a new family of hyperplane arrangements in dimension n ≥ 3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the ...
Rui Duarte, Antonio Guedes de Oliveira
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A Survey of the Shi Arrangement
Association for Women in Mathematics Series, 2019 I would like to keep a running version on my website. If you know of work on the Shi arrangement that I have missed, please send me a tex summary and references.
Susanna Fishel
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