Dual Artin Groups Explored

• Dual Artin groups are defined from Coxeter systems via reflection factorizations and noncrossing partition posets, offering a dual presentation that parallels standard Artin groups.
• The defining relations leverage the Hurwitz action and interval group methods to construct Garside structures, facilitating solutions to word and conjugacy problems in cases like affine and rank-three types.
• Recent extensions address standard–dual isomorphism challenges, explore novel interval Garside structures, and identify open problems in higher-rank non-spherical settings.

Dual Artin groups are groups defined from a Coxeter system by replacing the standard generating set of simple reflections with generators indexed by reflections that occur in reduced factorizations of a chosen Coxeter element, and by encoding relations through the corresponding noncrossing partition poset, interval group, or Hurwitz group. In spherical type, in affine type, in all rank-three Coxeter groups, and for Artin groups of XXL type, the dual group is known to be isomorphic to the corresponding standard Artin group; in higher-rank non-spherical, non-affine types the general standard–dual isomorphism problem remains open (Paolini et al., 30 Aug 2025, Delucchi et al., 2022, O'Brien, 11 Jun 2026).

1. Standard and dual constructions

Let $(W,S)$ be a Coxeter system. The standard Artin group is

$$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$

If $R\subset W$ is the set of all reflections and $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$ is a Coxeter element, the noncrossing partition poset

$\NC(W,w)=[1,w]$

is the interval in reflection-length order, and the dual Artin group $G^*_{W,w}$ is generated by symbols $\{r\}$ for $r\in R_0$, where $R_0$ is the subset of reflections that actually occur as labels along geodesics from $1$ to $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$0. Its defining relations identify the products attached to any two maximal chains from $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$1 to $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$2 in $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$3 (Paolini et al., 30 Aug 2025).

A second formulation uses interval groups. If $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$4 is viewed with generating set $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$5, then for a Coxeter element $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$6 the interval $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$7 in the reflection Cayley graph determines an interval group $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$8, generated by the reflections that occur on the interval and with relations given by loops in the Hasse diagram of $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$9. In finite or affine type, the natural homomorphism $R\subset W$0 is an isomorphism (Paolini et al., 2019).

A third formulation uses Hurwitz groups. If $R\subset W$1 is a Coxeter element in a Coxeter group $R\subset W$2, then the dual Artin group can be defined as

$R\subset W$3

where $R\subset W$4 is the Hurwitz group attached to the braid-group orbit of the tuple $R\subset W$5. In this description the dual Artin group also admits a presentation in terms of the set $R\subset W$6 of Hurwitz words in the free group $R\subset W$7 (O'Brien, 11 Jun 2026).

2. Hurwitz action, reflection factorizations, and canonical maps

The braid group acts on factorizations by the Hurwitz action: $R\subset W$8 and this action preserves the total product. For dual Artin groups, the Hurwitz action organizes reduced reflection factorizations of the Coxeter element and of its divisors, and it converts the chain relations in $R\subset W$9 into local two-term relations (Resteghini, 2024).

In the reflection-based presentation, one has the Hurwitz presentation

$w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$0

There is a canonical surjective morphism

$w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$1

and Resteghini isolates two structural conditions under which $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$2 is an isomorphism: the tuple $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$3 must be well-stabilized, meaning $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$4, and the triple $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$5 must be pan-transitive, meaning that the Hurwitz action on $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$6 is transitive for every $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$7 (Resteghini, 2024).

The Hurwitz-group formulation yields a different criterion. Let $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$8 be the canonical epimorphism and let $w=s_{\sigma(1)}\cdots s_{\sigma(|S|)}$9 be the set of Hurwitz words. Then the canonical epimorphism from the standard to the dual Artin group is an isomorphism if and only if the projection $\NC(W,w)=[1,w]$0 is injective on the image of $\NC(W,w)=[1,w]$1; equivalently,

$\NC(W,w)=[1,w]$2

This criterion is the basis of the XXL-type theorem (O'Brien, 11 Jun 2026).

3. Garside structures and the lattice property

Dual Artin groups are most rigid when the interval $\NC(W,w)=[1,w]$3 is a balanced lattice. In that case the associated interval monoid is a Garside monoid and the interval group is a Garside group, so the dual presentation supplies normal forms and the usual Garside-theoretic control of word and conjugacy problems (Neaime, 2019).

In spherical type, this lattice property is the classical source of the dual braid monoid. In affine type, the situation is sharply more selective. There exist dual interval Garside structures for the affine Artin groups of type $\NC(W,w)=[1,w]$4, $\NC(W,w)=[1,w]$5, and $\NC(W,w)=[1,w]$6. For all the other affine Artin groups, the intervals corresponding to Coxeter elements are not lattices and thus do not give rise to dual interval Garside structures (Neaime, 2019).

McCammond gives a more refined Euclidean classification. The unique dual presentation of $\NC(W,w)=[1,w]$7 is a Garside structure when $\NC(W,w)=[1,w]$8 is $\NC(W,w)=[1,w]$9 or $G^*_{W,w}$0, and it is not a Garside structure when $G^*_{W,w}$1 is $G^*_{W,w}$2, $G^*_{W,w}$3, $G^*_{W,w}$4, or $G^*_{W,w}$5. When the group has type $G^*_{W,w}$6 there are distinct dual presentations and the one investigated by Digne is the only one that is a Garside structure. The obstruction is the failure of the lattice property in the factorization poset $G^*_{W,w}$7, witnessed by bowties; a general criterion states that if the horizontal root system $G^*_{W,w}$8 is reducible, then $G^*_{W,w}$9 contains a bowtie and is not a lattice (McCammond, 2013).

This distinction is conceptually important. The existence of a dual Artin group is far broader than the existence of a dual Garside structure: the group-theoretic standard–dual isomorphism may hold even when the noncrossing interval fails to be a lattice.

4. The affine $\{r\}$0 model case

François Digne’s construction for affine type $\{r\}$1 is a model example of a dual Garside structure in infinite type. Let $\{r\}$2 be the Artin–Tits group of affine type $\{r\}$3. Digne proves that $\{r\}$4 is the group of fractions $\{r\}$5 of a Garside monoid $\{r\}$6, constructed via the generated group method from the Coxeter group $\{r\}$7 and a Coxeter element $\{r\}$8. The dual presentation is

$\{r\}$9

and the corresponding group presentation defines the dual Artin group. A key point is that $r\in R_0$0 is realized as a fixed-point subgroup in $r\in R_0$1 under an involution, and $r\in R_0$2 is the fixed-point germ inside the dual Garside germ for type $r\in R_0$3 (Digne, 2010).

The construction depends on a transitivity theorem for the Hurwitz action: if $r\in R_0$4 is a Coxeter element of $r\in R_0$5 and $r\in R_0$6, then the Hurwitz action of the braid group on the set of reduced decompositions of $r\in R_0$7 into reflections is transitive. This transitivity allows all relations in the dual monoid to be generated by the local Hurwitz relations above. The resulting Garside structure yields normal forms and a solution to the word problem in $r\in R_0$8 (Digne, 2010).

Digne also derives structural consequences for periodic elements. If $r\in R_0$9 is the lift of the Coxeter element, then for any integer $R_0$0, the centralizer $R_0$1 is isomorphic to the Artin–Tits group of type $R_0$2. The paper also states that the center of $R_0$3 is trivial (Digne, 2010).

5. Topological models and rank-three extension

In the affine case, dual Artin groups are central to the proof of the $R_0$4 conjecture. Paolini and Salvetti attach to $R_0$5 an interval complex $R_0$6, whose fundamental group is the dual Artin group $R_0$7, and a subcomplex

$R_0$8

that is homotopy equivalent to the orbit configuration space $R_0$9. For affine $1$0, there is a deformation retraction

$1$1

so

$1$2

This yields both the standard–dual isomorphism in affine type and the affine $1$3 theorem. The proof uses discrete Morse theory, EL-shellability of affine noncrossing partition posets, and the axial order on reflections (Paolini et al., 30 Aug 2025).

Rank three is the first genuinely non-affine infinite setting in which the dual program is complete. For every rank-three Coxeter system, the noncrossing partition poset $1$4 is a lattice and is EL-shellable, every element $1$5 is a Coxeter element for the subgroup generated by reflections $1$6, and the dual Artin group $1$7 is naturally isomorphic to the standard Artin group $1$8. Within this framework, one obtains a dual Garside structure, a proof of the $1$9 conjecture, the triviality of the center in the non-spherical cases, and the solubility of the word problem for rank-three Artin groups (Delucchi et al., 2022).

These topological results clarify the role of dual Artin groups. The dual interval complex is not merely a presentation device; in affine type and rank three it is a classifying-space model whose cells are governed directly by noncrossing partitions and reflection factorizations.

6. Recent extensions, alternatives, and open problems

The current frontier combines abstract criteria, new positive families, and a clearer understanding of what duality does not automatically provide. Resteghini proves that if two Coxeter systems are well-stabilized and pan-transitive, then their free product or direct product, with a suitable Coxeter element, again satisfies those two conditions; consequently the canonical morphism $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$00 is an isomorphism for the product system. Independently, the XXL-type theorem shows that if all defining integers satisfy $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$01, then for any Coxeter element $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$02 the dual Artin group $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$03 is canonically isomorphic to the standard Artin group $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$04 (Resteghini, 2024, O'Brien, 11 Jun 2026).

At the same time, failure of the lattice property has redirected attention toward interval Garside structures not related to dual presentations. For affine type $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$05, Neaime constructs interval Garside structures from the Shi and Corran–Lee–Lee presentations rather than from the reflection-based dual presentation, describing this as the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups (Neaime, 2019).

The main open questions remain those isolated by Paolini and Salvetti. Is every dual Artin group $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$06 isomorphic to the corresponding standard Artin group $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$07? Is the interval complex $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$08 a $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$09 for all $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$10? Is $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$11 homotopy equivalent to, or a deformation retract of, $$G_W \;=\; \left\langle S \,\middle|\, \underbrace{stst\cdots}_{m(s,t)\ \text{letters}} = \underbrace{tsts\cdots}_{m(s,t)\ \text{letters}} \ \forall s,t\in S \right\rangle.$$12 for all Coxeter groups? The known positive cases are spherical, affine, all rank-three Coxeter groups, and the XXL family, while the general higher-rank non-spherical, non-affine case remains open (Paolini et al., 30 Aug 2025).