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

Dyer, Matthew; Hohlweg, Christophe

doi:10.1016/j.aim.2016.06.022

Cited by 20 publications

(59 citation statements)

References 41 publications

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“…However, this cannot happen in an Artin-Tits monoid: Proposition 2.30. [12,20] A (finitely generated) Artin-Tits monoid contains finitely many basic elements.…”

Section: 3mentioning

confidence: 99%

Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Dehornoy¹

2017

J. Comb. Algebra

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Abstract. We describe a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group U (M ) of a monoid M in which any two elements admit a greatest common divisor. The method relies on a rewrite system R M that extends free reduction for free groups. Here we show that, if M satisfies what we call the 3-Ore condition about common multiples, what corresponds to type FC in the case of Artin-Tits monoids, then the system R M is convergent. Under this assumption, we obtain a unique representation result for the elements of U (M ), extending Ore's theorem for groups of fractions and leading to a solution of the word problem of a new type. We also show that there exist universal shapes for the van Kampen diagrams of the words representing 1.

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“…However, this cannot happen in an Artin-Tits monoid: Proposition 2.30. [12,20] A (finitely generated) Artin-Tits monoid contains finitely many basic elements.…”

Section: 3mentioning

confidence: 99%

Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Dehornoy¹

2017

J. Comb. Algebra

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“…A 0-low element is called a low element and we write L(W ) := L 0 (W ). Low elements were introduced by P. Dehornoy, M. Dyer, and the first author in [11], and extended for any n ∈ N by M. Dyer and the first author in [13]. We refer the reader to [13, §3.1- §3.3] for more details and examples; examples of low elements are also given in Figure 4.…”

Section: Small Inversion Sets and Low Elementsmentioning

confidence: 99%

“…A Garside shadow in (W, S) is a subset B ⊆ W that contains S and is closed under join (for the right weak order) and by taking suffixes. In [13], the authors show that finite Garside shadows exist in any Coxeter system (W, S). Let B be a finite Garside shadow in (W, S).…”

Section: Introductionmentioning

confidence: 99%

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Automata, reduced words and Garside shadows in Coxeter groups

Hohlweg¹,

Nadeau²,

Williams³

2016

Journal of Algebra

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Abstract. In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W, S). The definition of these automata is straightforward as it only requires the notion of weak order on (W, S) and the related notion of Garside shadows in (W, S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) the automata associated to the smallest Garside shadow is minimal; (2) the canonical automaton is minimal if and only if the support of all small roots is spherical, i.e., the corresponding root system is finite.

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“…The reasons for believing in Conjecture A are multiple. One abstract reason is that reduction is really specific and uses both the whole Garside structure of Artin-Tits monoids and, for the finiteness of the set of basic elements, some highly nontrivial properties of the associated Coxeter groups [15,21]: this may be seen as more promising than a generic approach based on, say, a "blind" Knuth-Bendix completion. Next, we state several related conjectures ("B", "C", "C unif "), of which some partial cases are proven and which suggest the existence of a rich rigid structure.…”

Section: Introductionmentioning

confidence: 99%

Multifraction reduction II: Conjectures for Artin–Tits groups

Dehornoy¹

2017

J. Comb. Algebra

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Multifraction reduction is a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system ("reduction") that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin-Tits monoids, and provide partial results and numerical evidence supporting such conjectures.Conjecture A. Reduction is semi-convergent for every Artin-Tits monoid.Proof. The argument is similar to the proof of [13, Proposition 2.4], and we only point the differences due to using signed multifractions. For (i), associativity is checked directly, and the generating subsets for F ± M and F M follow from the equalities (2.7) a 1 /···/a n = a 1 · a 2 · a 3 · a 4 · ··· = a 1 · 1/a 2 · a 3 · 1/a 4 · ··· :both hold in F ± M , and the second only involves positive multifractions. For (ii) and (iii), for every a in F ± M , the definition of ≃ ± implies ι(a) = ι(a) −1 and, writing ι + (a) for the ≃-class of a, that of ≃ implies ι + (1/a) = ι + (a) −1 . Hence both F ± M /≃ ± and F M /≃ are groups generated by M . One easily checks that the latter groups satisfy the universal property defining U(M ), and are therefore isomorphic to U(M ). Then (2.6) directly follows from (2.7).Next, for every a in F ± M , the product 1 · a belongs to F M . Then, for a, b in F ± M , write a ≈ b for 1 · a ≃ 1 · b. By considering all sign combinations and using relations like 1/ab ≃ 1/b · 1/a, However, a number of Artin-Tits monoids fail to be of type FC and therefore are not eligible for Proposition 2.19, typically the monoid of type A 2 considered in Example 2.16. So we are left with the question of either weakening the assumptions for Proposition 2.19, or using a conclusion weaker than convergence. Semi-convergenceAfter showing in Subsection 3.1 that the 3-Ore assumption cannot be weakened when proving the convergence of R ± M , we introduce in Subsection 3.2 a new property of R ± M called semiconvergence, which, as the name suggests, is weaker than convergence. We conjecture that, for every Artin-Tits monoid, the system R ± M is semi-convergent ("Conjecture A"). We prove in Subsection 3.3 that most of the consequences known to follow from the convergence of R ± M follow from its semi-convergence, in particular in terms of controlling the group U(M ) from inside the monoid M and solving its word problem. Finally, we describe in Subsection 3.4 several variants of semi-convergence.3.1. The strength of the 3-Ore condition. A first attempt for improving Proposition 2.19 could be to establish the convergence of R ± M from an assumption weaker than the 3-Ore condition. This approach fails, as the latter turns out to be not only sufficient, but also ...

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

Cited by 20 publications

References 41 publications

Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC

Automata, reduced words and Garside shadows in Coxeter groups

Multifraction reduction II: Conjectures for Artin–Tits groups

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