Complex of Irreducible Parabolic Subgroups

• The complex of irreducible parabolic subgroups is a flag simplicial complex built from proper irreducible parabolic subgroups whose vertices are defined by the commutation of canonical central elements.
• It relies on strong lattice properties where intersections remain parabolic and every element has a unique minimal parabolic closure, ensuring a well-structured ambient poset.
• Analogous to the curve complex in braid groups, this construction provides a unifying geometric framework with promising insights into hyperbolicity and large-scale group geometry.

The complex of irreducible parabolic subgroups is a simplicial complex attached most explicitly to Artin–Tits groups of spherical type and, more generally, to complex braid groups, in which vertices are proper irreducible parabolic subgroups and simplices encode pairwise compatibility through commuting distinguished central elements (Cumplido et al., 2017). In this setting, the construction depends on strong structural results about parabolic subgroups: intersections are again parabolic, the set of parabolic subgroups forms a lattice under inclusion, and every element admits a unique minimal parabolic subgroup containing it, called its parabolic closure (Cumplido et al., 2017). Closely related notions also occur in Coxeter groups, FC type Artin groups, and several algebraic-group settings, although in those cases the phrase often denotes an analogue or an interpretive framework rather than exactly the same simplicial object (Nuida, 2012).

1. Definition in Artin–Tits groups of spherical type

Let $A_S$ be an Artin–Tits group of spherical type, so the associated Coxeter group $W_S$ is finite. For each subset $X \subseteq S$, the subgroup $A_X$ generated by $X$ is a standard parabolic subgroup, and any conjugate $\alpha A_X \alpha^{-1}$ is a parabolic subgroup. Such a parabolic subgroup is irreducible when the induced Coxeter graph $\Gamma_X$ is connected (Cumplido et al., 2017).

The construction of the complex uses a canonical central element attached to each parabolic subgroup. If $P=A_X$ is standard, with Garside element $\Delta_X$, define

$$z_X = \begin{cases} \Delta_X, & \text{if } \Delta_X \text{ is central in } A_X,\ \Delta_X^2, & \text{otherwise}. \end{cases}$$

If $W_S$0, then

$W_S$1

This element is independent of the chosen representative of $W_S$2, and conjugation of parabolic subgroups is equivalent to conjugation of these distinguished central elements (Cumplido et al., 2017).

The complex of irreducible parabolic subgroups is then defined as the flag complex whose vertices are the proper irreducible parabolic subgroups of $W_S$3, and where a finite set $W_S$4 spans a simplex exactly when the associated central elements commute pairwise: $W_S$5 Equivalently, the $W_S$6-skeleton is the graph whose vertices are proper irreducible parabolics and whose edges join pairs with commuting distinguished centers (Cumplido et al., 2017).

A central structural theorem identifies this commutation criterion with a geometric compatibility relation. For distinct irreducible parabolic subgroups $W_S$7, the elements $W_S$8 and $W_S$9 commute if and only if one of the following holds: $X \subseteq S$0, $X \subseteq S$1, or $X \subseteq S$2 and every element of $X \subseteq S$3 commutes with every element of $X \subseteq S$4 (Cumplido et al., 2017). Thus the complex simultaneously records nesting and orthogonality.

2. Parabolic closures, intersections, and the lattice of parabolics

The definition above is viable only because parabolic subgroups in spherical type Artin–Tits groups have unusually robust closure properties. The intersection of two parabolic subgroups is again parabolic, and the set of parabolic subgroups forms a lattice under inclusion (Cumplido et al., 2017). This gives a well-behaved ambient poset from which the simplicial complex is extracted.

A key ingredient is the parabolic closure of an element. For $X \subseteq S$5, there exists a unique minimal parabolic subgroup $X \subseteq S$6 containing $X \subseteq S$7, and it can be described as the intersection of all parabolic subgroups containing $X \subseteq S$8: $X \subseteq S$9 This subgroup is stable under powers,

$A_X$0

and therefore roots of an element lie in the same parabolic closure as the element itself (Cumplido et al., 2017).

These properties have two immediate consequences for the complex. First, irreducible parabolics behave as canonical supports for elements and subgroups. Second, the compatibility relation among vertices can be interpreted inside a genuine lattice of parabolic subgroups rather than merely inside a set of conjugacy classes. In braid-type examples this recovers the familiar incidence behavior of embedded subsurfaces or nested supports (Cumplido et al., 2017).

The same pattern persists, in weaker form, beyond spherical type. For Artin groups of FC type, the class of finite type parabolic subgroups is closed under intersection, and the minimal finite type parabolic subgroup containing an element exists whenever the element belongs to a finite type parabolic subgroup (Morris-Wright, 2019). This allows an FC type analogue of the complex to be defined using finite type proper irreducible parabolic subgroups as vertices (Morris-Wright, 2019).

3. Relation to the curve complex

The motivating analogy is with the curve complex of a surface. In the $A_X$1-strand braid group $A_X$2, irreducible parabolic subgroups correspond to mapping classes supported on subdisks, and the distinguished central element $A_X$3 corresponds essentially to a Dehn twist about the associated curve or to a conjugate of a standard generator in the two-puncture case (Cumplido et al., 2017). Under this correspondence, commuting of central elements translates into disjointness or nesting of curves.

Accordingly, in braid groups the complex of irreducible parabolic subgroups recovers the curve complex of the punctured disk (Cumplido et al., 2017). The construction is therefore not merely analogous to the curve complex; in type $A_X$4 it is a reformulation of it in purely group-theoretic terms.

This perspective was one of the principal motivations for defining the complex in the first place. The complex supports a simplicial action of the Artin–Tits group by conjugation, and this action is by isometries for the graph metric on the $A_X$5-skeleton (Cumplido et al., 2017). Because the complex is a flag complex, its large-scale geometry is already determined by the graph of pairwise commuting distinguished central elements.

The hyperbolicity question is also modeled on the curve-complex paradigm. For spherical type Artin–Tits groups, it was conjectured that the complex of irreducible parabolic subgroups is $A_X$6-hyperbolic (Cumplido et al., 2017). The same conjectural picture later reappeared in the settings of FC type Artin groups and complex braid groups (Morris-Wright, 2019).

4. Extensions to FC type Artin groups and complex braid groups

The construction admits two major generalizations: to FC type Artin groups and to generalized braid groups of complex reflection groups.

In FC type Artin groups, the relevant complex is built from finite type proper irreducible parabolic subgroups. Adjacency is defined exactly as in the spherical case: two such subgroups are adjacent if one is properly contained in the other, or if they intersect trivially and commute elementwise (Morris-Wright, 2019). Morris-Wright shows that this adjacency is equivalent to commutation of the associated central elements $A_X$7, and that the resulting simplicial complex is a flag complex (Morris-Wright, 2019). For irreducible FC type Artin groups whose defining graph is not a join and has at least two vertices, this complex has infinite diameter; more precisely, if the defining graph is not a join and has at least two vertices, then the complex of parabolic subgroups has infinite diameter (Morris-Wright, 2019). The paper conjectures that for an irreducible non-cyclic FC type Artin group, the complex is an infinite diameter hyperbolic space (Morris-Wright, 2019).

For generalized braid groups $A_X$8 attached to irreducible complex reflection groups $A_X$9, Marin and González-Meneses define parabolic subgroups topologically via local fundamental groups near the discriminant, rather than through a chosen Artin presentation (González-Meneses et al., 2022). In that setting, the curve graph has vertices given by irreducible parabolic subgroups of $X$0, and two distinct vertices $X$1 are adjacent when

$X$2

The associated clique complex is the analogue of the complex of irreducible parabolic subgroups (González-Meneses et al., 2022).

As in the Artin–Tits case, each irreducible parabolic subgroup $X$3 carries a canonical central element $X$4, and one has

$X$5

where $X$6 denotes parabolic closure. Moreover, two irreducible parabolic subgroups are adjacent if and only if their distinguished central elements commute (González-Meneses et al., 2022). This gives a uniform algebraic characterization of the complex in terms of central generators.

The general theory originally excluded the exceptional group $X$7, because $X$8 had no known Garside group structure. Garnier completed the theory using a Garside groupoid. For $X$9, parabolic subgroups again form a lattice, every element has a unique parabolic closure $\alpha A_X \alpha^{-1}$0 with $\alpha A_X \alpha^{-1}$1, and adjacency of irreducible parabolic subgroups is characterized by commuting distinguished central elements exactly as in the earlier theory (Garnier, 2024). In addition, Garnier gives an explicit classification of parabolic subgroups of $\alpha A_X \alpha^{-1}$2 up to conjugacy (Garnier, 2024).

Setting	Vertices	Adjacency criterion
Spherical Artin–Tits group	Proper irreducible parabolic subgroups	Pairwise commutation of $\alpha A_X \alpha^{-1}$3 (Cumplido et al., 2017)
FC type Artin group	Finite type proper irreducible parabolic subgroups	Nesting or trivial commuting intersection; equivalently $\alpha A_X \alpha^{-1}$4-commutation (Morris-Wright, 2019)
Complex braid group $\alpha A_X \alpha^{-1}$5	Irreducible parabolic subgroups	Nesting or $\alpha A_X \alpha^{-1}$6; equivalently $\alpha A_X \alpha^{-1}$7-commutation (González-Meneses et al., 2022)

5. Coxeter-group and centralizer analogues

In Coxeter theory the phrase may refer not to the same simplicial construction, but to a related combinatorial object built from parabolic configurations and their centralizers. Nuida studies the centralizer $\alpha A_X \alpha^{-1}$8 of a parabolic subgroup $\alpha A_X \alpha^{-1}$9 in a Coxeter group and shows that it contains an inner factor $\Gamma_X$0 identified with the fundamental group of a $\Gamma_X$1-cell complex $\Gamma_X$2, while the reflection part is $\Gamma_X$3 (Nuida, 2012). In this setting, $\Gamma_X$4 can be regarded as a complex assembled from irreducible finite extensions of the base parabolic configuration, and the action of $\Gamma_X$5 on finite irreducible components of $\Gamma_X$6 is trivial when $\Gamma_X$7 is $\Gamma_X$8-free (Nuida, 2012).

For Coxeter groups of arbitrary rank, ordinary parabolic closures need not be parabolic. Nuida therefore introduces locally parabolic subgroups and locally parabolic closures, proving that arbitrary intersections of locally parabolic subgroups remain locally parabolic (Nuida, 2010). This result suggests that in infinite-rank Coxeter settings the natural ambient category for a parabolic complex is larger than the class of genuine parabolics (Nuida, 2010).

There is also a direct Artin-group analogue. Godelle, Paris, and collaborators study complete parabolic subgroups of general Artin groups and use a CAT(0) clique-cube complex $\Gamma_X$9, whose vertices are cosets $P=A_X$0 with $P=A_X$1 complete or empty, ordered by inclusion (Möller et al., 2022). Vertex stabilizers are complete parabolic subgroups, and the complex provides a generalized Deligne complex that organizes these “complete” parabolic pieces (Möller et al., 2022). Although the paper does not use the phrase “complex of irreducible parabolic subgroups,” complete parabolics play that structural role.

6. Algebraic-group and representation-theoretic interpretations

Outside braid and Artin settings, the phrase often appears in an interpretive or analogical sense. In the geometry of nilradicals of parabolic subgroups of $P=A_X$2, the complement of the Richardson orbit decomposes into finitely many irreducible components indexed by a set $P=A_X$3, each given either by rank conditions $P=A_X$4 or by minimal tableau degenerations $P=A_X$5 (Baur et al., 2010). That work explicitly suggests viewing these irreducible boundary components as a “complex of irreducible degenerations” attached to a parabolic subgroup, with the Richardson orbit as the open cell and the $P=A_X$6 as primitive boundary faces (Baur et al., 2010).

In the theory of exceptional algebraic groups, “irreducible” means not contained in any proper parabolic subgroup, following Serre’s definition (Thomas, 2015). Thomas’s classification of irreducible $P=A_X$7 subgroups of exceptional groups and the later classification of irreducible connected subgroups provide explicit finite lattices of overgroups for such irreducible subgroups (Thomas, 2015, Thomas, 2016). These lattices are not the same as the Artin-group complex, but they supply a poset-theoretic analogue in which incidence is governed by containment among irreducible subgroups and reductive overgroups rather than by commuting distinguished central elements.

A related but distinct usage occurs in commuting varieties of parabolic subalgebras. Goddard and Goodwin characterize irreducibility of the commuting variety $P=A_X$8 in terms of the modality of the $P=A_X$9-action on $\Delta_X$0, and for Borels they classify exactly when $\Delta_X$1 is irreducible and when it is normal (Goddard et al., 2016). Here the relevant “complex” is the subposet of parabolics whose commuting varieties remain irreducible or normal, rather than a graph defined by commuting centers.

In these algebraic-group contexts, therefore, the term denotes a family of related constructions: lattices of irreducible subgroups, boundary complexes of parabolic orbit geometry, and incidence structures governed by parabolic containment. A plausible implication is that the Artin-group complex sits within a broader pattern in which parabolic subgroup data are organized by either compatibility relations or irreducible degenerations, but only in the Artin–Tits and braid-group settings is the phrase realized as a single canonical flag complex (Cumplido et al., 2017, González-Meneses et al., 2022).

7. Conceptual significance and current scope

The complex of irreducible parabolic subgroups is most sharply defined in three settings: spherical Artin–Tits groups, FC type Artin groups, and generalized braid groups of complex reflection groups (Cumplido et al., 2017, Morris-Wright, 2019, González-Meneses et al., 2022). In each case, the construction relies on three structural facts: parabolic subgroups admit a workable intersection theory, elements possess canonical parabolic closures, and irreducible parabolics are detected by distinguished central elements whose commutation encodes compatibility.

This makes the complex a unifying geometric object for groups that otherwise lack a surface model. In braid groups it recovers the curve complex of the punctured disk; in spherical Artin–Tits groups and complex braid groups it serves as a proposed replacement for the curve complex; and in FC type Artin groups it is intended to capture analogous large-scale geometry even when finite-type methods no longer apply (Cumplido et al., 2017, Morris-Wright, 2019). The conjectural $\Delta_X$2-hyperbolicity of these complexes remains one of the central open directions in the subject (Cumplido et al., 2017, González-Meneses et al., 2022).

At the same time, the broader literature shows that “complex of irreducible parabolic subgroups” is not a uniquely fixed phrase across all fields. In Coxeter theory, algebraic-group theory, and representation theory, it often designates a related incidence structure attached to parabolics, irreducible centralizers, or irreducible degenerations rather than the specific flag complex defined by commuting parabolic centers [(Nuida, 2012); (Baur et al., 2010); (Thomas, 2016)]. The most precise encyclopedic use therefore identifies the term with the flag complex on irreducible parabolic subgroups in Artin–Tits and complex braid settings, while recognizing that several nearby theories provide natural analogues.