A Survey of the Shi Arrangement

Fishel, Susanna

doi:10.1007/978-3-030-05141-9_3

Cited by 6 publications

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References 69 publications

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“…In [40,41], Shi introduced the Shi arrangement Shi(𝑊, 𝑆) = Shi 0 (𝑊, 𝑆) in the case of irreducible affine Weyl groups to study Kazhdan-Lusztig cells for 𝑊. Several surprising connections with Shi arrangements for affine Weyl groups have been studied, from ad-nilpotent ideals of Borel subalgebras [11] to Catalan arrangements [3,5]; see also [27] and the references within. In [41], Shi proves a conjecture by Carter on the number of sign-types of an affine Weyl group.…”

Section: Introductionmentioning

confidence: 99%

Shi arrangements and low elements in Coxeter groups

Dyer,

Fishel,

Hohlweg

et al. 2024

Proceedings of London Math Soc

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Given an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

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Section: Introductionmentioning

confidence: 99%

Shi arrangements and low elements in Coxeter groups

Dyer,

Fishel,

Hohlweg

et al. 2024

Proceedings of London Math Soc

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“…The Shi arrangement of (W, S) is a finite hyperplane subarrangement of A(W, S), which was introduced by Shi in [30,Chapter 7] to study Kazhdan-Lusztig cells in affine Weyl groups of type Ã and later extended to any affine Weyl groups in [32]; see also Fishel's survey [21]. In these affine cases, the Coxeter arrangement and the Shi arrangement are naturally geometrically realized as affine hyperplane arrangements in a Euclidean vector space; examples are given in Figure 1, Figure 2 and Figure 6.…”

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confidence: 99%

Shi arrangements and low elements in affine Coxeter groups

Chapelier-Laget¹,

Hohlweg²

2022

Preprint

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Given an affine Coxeter group W , the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for W . In particular, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W . Low elements in W were introduced to study the word problem of the corresponding Artin-Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W .In this article, we show in the case of an affine Coxeter group that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a byproduct of our proof, we show that the descent-walls -the walls that separate a region from the fundamental alcove -of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element. Contents 14 5. Shi's admissible signs and Shi-regions 16 6. Smallest elements in Shi-regions 23 7. Descent-roots and descent-walls of Shi-regions 25 8. Minimal elements in Shi-regions are low 32 References 34 N (w) = Φ + ∩ w(Φ − ),

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Shi Arrangements Restricted to Weyl Cones

Dorpalen-Barry,

Stump

2024

Ann. Comb.

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A Survey of the Shi Arrangement

Cited by 6 publications

References 69 publications

Shi arrangements and low elements in Coxeter groups

Shi arrangements and low elements in Coxeter groups

Shi arrangements and low elements in affine Coxeter groups

Shi Arrangements Restricted to Weyl Cones

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