Garside Monoids Research Articles

Odd and even Fibonacci lattices arising from a Garside monoid

Odd and even Fibonacci lattices arising from a Garside monoid

A new Garside structure on torus knot groups and some complex braid groups

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For [Formula: see text] two coprime integers, we introduce a new Garside monoid [Formula: see text] having as Garside group the [Formula: see text]-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the [Formula: see text]-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for [Formula: see text] and for dihedral Artin groups of even type.

A note on Garside monoids and $${\mathscr {M}}$$-braces

A note on Garside monoids and $${\mathscr {M}}$$-braces

RETRACTED ARTICLE: A note on Garside monoids and $$\mathscr {M}$$-braces

RETRACTED ARTICLE: A note on Garside monoids and $$\mathscr {M}$$-braces

On the growth of Artin–Tits monoids and the partial theta function

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin–Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix.Using this approach, we show that the exponential growth rates of the Artin–Tits monoids of type An (positive braid monoids) tend to 3.233636… as n tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only formal power series x0(y)=−(1+y+2y2+4y3+9y4+⋯) which is the leading root of the classical partial theta function.We also describe the sequence 1,1,2,4,9,… formed by the coefficients of −x0(y), by showing that its kth term (the coefficient of yk) is equal to the number of braids of length k, in the positive braid monoid A∞ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator a1. This is an unexpected connection between the partial theta function and the theory of braids.

On some torus knot groups and submonoids of the braid groups

The submonoid of the 3-strand braid group B3 generated by σ1 and σ1σ2 is known to yield an exotic Garside structure on B3. We introduce and study an infinite family (Mn)n≥1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M2 is isomorphic to the above monoid. The corresponding Garside group G(Mn) is isomorphic to the (n,n+1)-torus knot group–which is isomorphic to B3 for n=2 and to the braid group of the exceptional complex reflection group G12 for n=3. This yields a new Garside structure on (n,n+1)-torus knot groups, which already admit several distinct Garside structures.The (n,n+1)-torus knot group is an extension of Bn+1, and the Garside monoid Mn surjects onto the submonoid Σn of Bn+1 generated by σ1,σ1σ2,…,σ1σ2⋯σn, which is not a Garside monoid when n>2. Using a new presentation of Bn+1 that is similar to the presentation of G(Mn), we nevertheless check that Σn is an Ore monoid with group of fractions isomorphic to Bn+1, and give a conjectural presentation of it, similar to the defining presentation of Mn. This partially answers a question of Dehornoy–Digne–Godelle–Krammer–Michel.

Braid groups of type ADE, Garside monoids, and the categorified root lattice

We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.

Decomposition of Garside groups and self-similar L-algebras

Picantin's iterated crossed product representation of Garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure. It is shown that the tree-like structure of such an iterated crossed product is equivalent to a partial cycle set, closely related to a class of set-theoretic solutions of the quantum Yang–Baxter equation. The decomposition of finite square-free solutions is related to the crossed product representation of the corresponding structure group.

On the penetration distance in Garside monoids

We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irreducible Artin monoids of spherical type, extending a result by Caruso.Our results establish that the expected value of the penetration distance pd(x,y) in an irreducible Artin monoid of spherical type is bounded independently of the length of x, if x is chosen uniformly among all elements of given canonical length and y is chosen uniformly among all atoms; the latter in particular explains observations made by Thurston in the context of the braid group, and it shows that all irreducible Artin monoids of spherical type exhibit an analogous behaviour. Our results also give an affirmative answer to a question posed by Dehornoy.

Zappa–Szép products of Garside monoids

A monoid K is the internal Zappa–Szép product of two submonoids, if every element of K admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving. We prove that K is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of K and the Garside structure of K can be described in terms of the Garside structures of the factors. We give explicit isomorphisms between the lattice structures of K and the product of the lattice structures on the factors that respect the Garside normal forms. In particular, we obtain explicit natural bijections between the normal form language of K and the product of the normal form languages of its factors.

Fixed Points of Endomorphisms and Metrics in preGarside Monoids

E. Rodaro and P. V. Silva proved recently that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the particular case of Artin–Tits monoids, these submonoids are Artin–Tits monoids too. We prove algebraically some inequalities, equivalences, and nonequivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.

A half-factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on $${\mathbb {N}}^{*}$$ N ∗

The half-factorial, locally right Garside monoid of cofinite monotone sequences on \({\mathbb {N}}^{*}\) is considered in this paper. This is a right cancellative, left rigid Mobius monoid, and this monoid directs us to the inverse monoid of cofinite monotone partial bijections on \({\mathbb {N}}^{*}\). In a certain sense, the polycyclic inverse monoid is a 0-analogue of this derived inverse monoid. Similarities and differences between these two inverse monoids are presented involving Mobius categories, Leech’s construction, breaking processes, gauge inverse submonoids, etc.

Garside families and Garside germs

Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various criteria for establishing the existence of normal decompositions of a certain type.

Garside Structures on Monoids with Quadratic Square-Free Relations

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra \(A= \textbf{k}S\) has a Frobenius Koszul dualA! with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.

A Garside presentation for Artin-Tits groups of type \widetilde{C}_n

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the ''generated group'' method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde C$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type $\tilde A$ under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

Growth function for a class of monoids

In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras. Nous étudions une classe de monoïdes incluant les monoïdes de Garside, et donnons une preuve combinatoire simple d'une formule pour la somme formelle de leurs éléments. Cela mène à une formule pour la fonction de croissance du monoïde dans le cas homogène, et peut être aussi relevé en une résolution de l'algèbre de monoïdes. Ces résultats sont ensuite appliqués aux monoïdes liés aux systèmes de Coxeter : nous donnons la fonction de croissance des monoïdes d'Artin-Tits ainsi que des monoïdes duaux ; pour ces derniers nous montrons que leur algèbre de monoïde en types A et B est une algèbre de Koszul.

Alternating normal forms for braids and locally Garside monoids

We describe new types of normal forms for braid monoids, Artin–Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties (“locally Garside monoids”). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard order (“Dehornoy order”) of B n in terms of that of B n − 1 , and to a quadratic upper bound for the complexity of this order.

Parabolic subgroups of Garside groups

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains all Artin–Tits groups of spherical type. We show that the well-known notion of a parabolic subgroup of an Artin–Tits group can be extended to the framework of Garside groups so that most of the standard properties known in the Artin–Tits groups case are preserved. The extension is not trivial and it requires a new approach. We also define the more general notion of a Garside subgroup of a Garside group that nicely extends the notion of an LCM-homomorphism between Artin–Tits groups.

Présentations duales des groupes de tresses de type affine $Ã$

Artin–Tits groups of spherical type have two well-known Garside structures, coming respectively from the divisibility properties of the classical Artin monoid and of the dual monoid. For general Artin–Tits groups, the classical monoids have no such Garside property. In the present paper we define dual monoids for all Artin–Tits groups and we prove that for the type \tilde A_n we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin–Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin–Tits group. More precisely, if W is a Coxeter group, one can consider the length l_R on W with respect to the generating set R consisting of all reflections. Let c be a Coxeter element in W and let P_c be the set of elements p\in W such that c can be written c=pp' with l_R(c)=l_R(p)+l_R(p') . We define the monoid M(P_c) to be the monoid generated by a set \underline P_c in one-to-one correspondence, p\mapsto \underline p , with P_c with only relations \underline{pp'}=\underline p.\underline p' whenever p , p' and pp' are in P_c and l_R(pp')=l_R(p)+l_R(p') . We conjecture that the group of quotients of M(P_c) is the Artin–Tits group associated to W and that it has a simple presentation (see 1.1 (ii)). These conjectures are known to be true for spherical type Artin–Tits groups. Here we prove them for Artin–Tits groups of type \tilde A . Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type A .

A dual braid monoid for the free group

We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman–Ko–Lee and by the author of new Garside monoids for Artin groups of spherical type.