On Floors and Ceilings of the $k$-Catalan Arrangement

Thiel, Marco

doi:10.37236/4121

Cited by 4 publications

(9 citation statements)

References 6 publications

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“…Sulzgruber [Sul15] built on [GMV16] by finding the coordinates of the M -stable permutations, generalizing (3.5). Thiel [Thi14] extended their work to other types, among other results.…”

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confidence: 88%

A Survey of the Shi Arrangement

Fishel

2019

Association for Women in Mathematics Series

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In [Shi86], Shi proved Lusztig's conjecture that the number of two-sided cells for the affine Weyl group of type A n−1 is the number of partitions of n. As a byproduct, he introduced the Shi arrangement of hyperplanes and found a few of its remarkable properties. The Shi arrangement has since become a central object in algebraic combinatorics. This article is intended to be a fairly gentle introduction to the Shi arrangement, intended for readers with a modest background in combinatorics, algebra, and Euclidean geometry. After background material in Section 1, this introduction to the arrangement will be by way of a discussion in Section 2 of how it arose, some of its marvelous enumerative properties in Section 3, and some of its surprising connections to algebra in Section 4. For some brief comments on recent extensions, see Section 5 and for an incomplete list of topics we left out, see Section 6.

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“…Sulzgruber [Sul15] built on [GMV16] by finding the coordinates of the M -stable permutations, generalizing (3.5). Thiel [Thi14] extended their work to other types, among other results.…”

mentioning

confidence: 88%

A Survey of the Shi Arrangement

Fishel

2019

Association for Women in Mathematics Series

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“…The key result is that, in this case, the Shi polyhedron S 0 is a dilatation of the fundamental chamber C by a factor of (h + 1); see [38,Lemma 8.5 and Corollary 8.6]. The same method is used by Thiel in his thesis [41,Chapter 5] to recover the formula in Eq. ( 9): write S m in term of the ribbons used in [38,Lemma 8.5] with k α = −(m + 1) and the 'm' in Shi's article replaced by (m + 1)h + 1; this implies that S m is a dilation of C by a factor of (m + 1)h + 1.…”

Section: Extended Shi Arrangements and Low Elementsmentioning

confidence: 98%

“…For an arbitrary extended Shi arrangement, this formula is a consequence of a result of Yoshinaga [46], see also [2,Theorem 5.1.16] for more details. In 2015, then Thiel [41,42] recovered this results by applying Shi's method. • For indefinite types, we do not have closed formulas; see Table 1 for some examples.…”

Section: Extended Shi Arrangements and Low Elementsmentioning

confidence: 99%

“…n , which was first proved by Yoshinaga [46] using techniques from representation theory. In his thesis [41,Ch. 4], Thiel gives a direct proof by extending Shi's result to any m in the case of affine Coxeter systems; see also [42, §11].…”

Section: Introductionmentioning

confidence: 99%

“…Acknowledgements. The authors are grateful to Christian Stump for pointing out to us the reference [41], and many thanks to Vic Reiner and Nathan Williams for useful discussion on m-Shi arrangement in affine types.…”

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confidence: 99%

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Small roots, low elements, and the weak order in Coxeter groups

Dyer

Hohlweg

2016

Advances in Mathematics

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In this article we provide a new finite class of elements in any Coxeter system (W, S) called low elements. They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W, S). Our first main result is to show that they form a Garside shadow in (W, S), i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated ArtinTits groups have a finite Garside family. This result was announced in a note with P. Dehornoy in Comptes rendus mathématiques [9] in which the present article was referred to under the following working title: Monotonicity of dominancedepth on root systems and applications. The proof is based on a fundamental property enjoyed by small roots and which is our second main result; the set of small root is bipodal. For a natural number n, we define similarly n-low elements from n-small roots and conjecture that the set of n-small roots Root poset Dominance order Small roots is bipodal, implying the set of n-low elements is a Garside shadow; we prove this conjecture for affine Coxeter groups and Coxeter groups whose graph is labelled by 3 and ∞. To prove the latter, we extend the root poset on positive roots to a weak order on the root system and define a Bruhat order on the root system, and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups.

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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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On Floors and Ceilings of the $k$-Catalan Arrangement

Cited by 4 publications

References 6 publications

A Survey of the Shi Arrangement

A Survey of the Shi Arrangement

Small roots, low elements, and the weak order in Coxeter groups

Shi arrangements and low elements in Coxeter groups

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