Results 1 to 10 of about 19,462 (147)
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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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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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 $
Christos Athanasiadis
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Freeness for restriction arrangements of the extended Shi and Catalan arrangements [PDF]
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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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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Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements [PDF]
Journal of Combinatorial Theory - Series A, 2007 13 ...
Thomas Zaslavsky, David Forge
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On the two variable distance enumerator of the Shi hyperplane arrangement [PDF]
European Journal of Combinatorics, 2008 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Sivaramakrishnan Sivasubramanian
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Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements [PDF]
Journal of Algebraic Combinatorics, 2015 In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each filter is a free subarrangement satisfying the dual-partition formula.
Takuro Abe +2 more
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A Class of Labeled Posets and the Shi Arrangement of Hyperplanes [PDF]
Journal of Combinatorial Theory - Series A, 1997 An unlabeled poset is \((C_2+ 1)\)-free if it does not contain the three element poset \(\{p,q,r\}\) where ...
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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