Complex Projective Reflection Groups

• Complex projective reflection groups are defined as finite subgroups of PGL(V) generated by projective images of pseudo-reflections, linking classic reflection groups with projective geometry.
• The theory utilizes the Shephard–Todd classification, invariant polynomial rings, and exact sequences to explore group structures and associated braid groups in both finite and hyperbolic settings.
• Applications span complex hyperbolic geometry, arithmetic reflection lattices, and derived categorical structures, providing deep insights into algebraic and topological symmetries.

Complex projective reflection groups are finite subgroups of projective linear groups generated by projective images of pseudo-reflections, together with a complementary complex-hyperbolic usage for discrete projective-unitary groups generated by complex reflections acting on complex hyperbolic space. In the finite setting they arise by projectivizing complex reflection groups $W \le GL(V)$ to $PW=W/Z(W)$; in the hyperbolic setting they are subgroups of $PU(n,1)$ generated by holomorphic elliptic isometries fixing complex hyperplanes. The subject therefore links the Shephard–Todd classification, polynomial invariant theory, braid groups of arrangement complements, and arithmetic reflection lattices (Garnier, 31 Jul 2025, Mark et al., 2021).

1. Foundational notions

Let $V$ be a finite-dimensional complex vector space. A reflection, or pseudo-reflection, is a finite-order linear automorphism $r \in GL(V)$ that fixes pointwise a hyperplane in $V$. Equivalently, $r$ has eigenvalue $1$ with multiplicity $\dim V-1$ and one eigenvalue $\neq 1$. In the rank-two formulation emphasized by Buchweitz–Faber–Ingalls, such an element can be written as

$PW=W/Z(W)$0

with $PW=W/Z(W)$1 the fixed hyperplane and $PW=W/Z(W)$2 (Buchweitz et al., 2018).

A finite subgroup $PW=W/Z(W)$3 generated by reflections is a complex reflection group. Its projectivization is the image $PW=W/Z(W)$4, and a finite subgroup $PW=W/Z(W)$5 is a projective reflection group precisely when it is generated by images of reflections; equivalently, there exists a reflection group $PW=W/Z(W)$6 with $PW=W/Z(W)$7. The projectivization fits into the exact sequence

$PW=W/Z(W)$8

where $PW=W/Z(W)$9, and for irreducible $PU(n,1)$0 this is the scalar center by Schur’s lemma (Garnier, 31 Jul 2025).

A second standard setting uses an indefinite Hermitian form $PU(n,1)$1 of signature $PU(n,1)$2 on $PU(n,1)$3. The complex hyperbolic space $PU(n,1)$4 is the projectivization of the negative cone

$PU(n,1)$5

and the full holomorphic isometry group is $PU(n,1)$6. In this context a complex reflection is a holomorphic elliptic isometry whose fixed-point locus is a totally geodesic complex $PU(n,1)$7-plane. With a normal vector $PU(n,1)$8 to the mirror and rotation angle $PU(n,1)$9, a standard matrix representative is

$V$0

whose projectivization lies in $V$1 (Mark et al., 2021).

2. Classification, projectivization, and model families

The finite linear theory is organized by the Shephard–Todd classification of irreducible complex reflection groups. If $V$2 has rank $V$3, then the invariant ring is a polynomial algebra

$V$4

with homogeneous generators of degrees $V$5. The same data determine $V$6, $V$7, and the Hilbert series

$V$8

Among irreducible groups, the primitive or exceptional cases are the Shephard–Todd groups $V$9 with $r \in GL(V)$0 (Bonnafé, 2018).

Projective reflection groups in the sense of Caselli are systematic scalar quotients of the infinite Shephard–Todd family. For parameters $r \in GL(V)$1 with $r \in GL(V)$2, $r \in GL(V)$3, and $r \in GL(V)$4, one defines

$r \in GL(V)$5

where $r \in GL(V)$6 is the scalar subgroup generated by $r \in GL(V)$7. This family contains all classical Weyl groups, all complex reflection groups of type $r \in GL(V)$8, and their projectivizations. It also carries a duality exchanging $r \in GL(V)$9 and $V$0: $V$1 Biagioli and Caselli developed descent-like and major-index-like statistics on these groups and related them to Hilbert series of diagonal invariant algebras (Biagioli et al., 2011).

In rank two the projective viewpoint acquires an especially rigid form. The identification $V$2, together with the double cover $V$3, yields a bijective correspondence between finite complex reflection groups of rank two and finite real reflection groups in $V$4. In the Clifford-theoretic formulation, $V$5 is identified with $V$6, $V$7 with $V$8, and a complex order-two reflection in $V$9 maps to a real reflection in $r$0. This rank-two “magic square” explains the dihedral and Platonic cases without appealing to a table lookup (Buchweitz et al., 2018).

3. Invariants, singular hypersurfaces, and projective geometry

The polynomial invariant theory of finite reflection groups produces projective hypersurfaces with large symmetry and often extreme singularity counts. If $r$1 are basic invariants and the space of invariants of degree $r$2 is two-dimensional, then one obtains an invariant pencil

$r$3

where $r$4 is a monomial in lower-degree invariants of total degree $r$5. Singular members are isolated parameter values $r$6, detected by solving $r$7 together with its affine partial derivatives. The group action forces singular points to occur in full $r$8-orbits, and for almost all irreducible singular members in the primitive cases the action is transitive on the singular locus (Bonnafé, 2018).

Group	Invariant hypersurface	Singularities
$r$9	degree-8 surface $1$0	$1$1 $1$2
$1$3	degree-8 $1$4; degree-12 $1$5	$1$6 nodes; $1$7 $1$8
$1$9	degree-24 surface $\dim V-1$0	$\dim V-1$1 $\dim V-1$2
$\dim V-1$3	degree-14 curve	$\dim V-1$4 cusps $\dim V-1$5

These constructions push known lower bounds close to Miyaoka’s upper bound

$\dim V-1$6

In particular, the primitive-group examples yield $\dim V-1$7, $\dim V-1$8, and $\dim V-1$9. The same framework also produces highly singular plane curves, including a degree-14 $\neq 1$0-invariant curve with $\neq 1$1 cusps and various $\neq 1$2, $\neq 1$3, and $\neq 1$4 examples with $\neq 1$5, $\neq 1$6, $\neq 1$7, $\neq 1$8, and $\neq 1$9 singularities (Bonnafé, 2018).

The paper on singular curves and surfaces does not use the discriminant divisor explicitly, but it places the invariant pencils in the geometry of the quotient map $PW=W/Z(W)$00 and the arrangement of reflecting hyperplanes. A plausible implication is that the geometry of special invariant hypersurfaces is best understood as a projective shadow of the orbit map and the reflection arrangement.

4. Braid groups and projective topology

For a finite reflection group $PW=W/Z(W)$01 with reflecting arrangement $PW=W/Z(W)$02, the regular locus is

$PW=W/Z(W)$03

Its fundamental groups define the pure braid group

$PW=W/Z(W)$04

and the braid group

$PW=W/Z(W)$05

fitting into the classical exact sequence

$PW=W/Z(W)$06

Projectivization introduces a subtlety: the naive complement $PW=W/Z(W)$07 may still contain points with nontrivial stabilizers coming from regular eigenspaces of noncentral regular elements (Garnier, 31 Jul 2025).

The correct projective domain is therefore the strongly regular locus

$PW=W/Z(W)$08

for $PW=W/Z(W)$09. One then defines the projective pure braid group and projective braid group by

$PW=W/Z(W)$10

Over $PW=W/Z(W)$11 there is a principal $PW=W/Z(W)$12-bundle

$PW=W/Z(W)$13

which yields an exact sequence

$PW=W/Z(W)$14

where $PW=W/Z(W)$15 is the scalar loop and generates the center $PW=W/Z(W)$16. The main theorem of Garrel’s 2025 note is that for a nontrivial irreducible projective reflection group $PW=W/Z(W)$17, if $PW=W/Z(W)$18 is the maximal reflection-group lift with $PW=W/Z(W)$19, then

$PW=W/Z(W)$20

This proves Shvartsman’s conjecture for all complex projective reflection groups and corrects a claim of Broué–Malle–Rouquier: the analogous exactness over the naive projective complement holds precisely when all regular elements of $PW=W/Z(W)$21 are central (Garnier, 31 Jul 2025).

5. Complex-hyperbolic and arithmetic reflection lattices

The hyperbolic branch of the theory concerns discrete groups in $PW=W/Z(W)$22 generated by complex reflections. In the arithmetic setting, a basic family is provided by the Picard modular groups $PW=W/Z(W)$23, where $PW=W/Z(W)$24 is the ring of integers of $PW=W/Z(W)$25. Working in the Siegel model defined by

$PW=W/Z(W)$26

Paupert and Wells exhibited explicit matrices

$PW=W/Z(W)$27

and used finite presentations plus Magma index computations to prove that $PW=W/Z(W)$28 is generated by complex reflections for $PW=W/Z(W)$29, while for $PW=W/Z(W)$30 it has an index $PW=W/Z(W)$31 subgroup generated by complex reflections. In the quaternionic setting, the Hurwitz modular group $PW=W/Z(W)$32 is generated by quaternionic reflections. The computational strategy is to add relations killing a reflection word and compute the resulting quotient order; in the quaternionic case the full presentation has $PW=W/Z(W)$33 generators and $PW=W/Z(W)$34 relations, so a truncated presentation with the first $PW=W/Z(W)$35 relations is used (Mark et al., 2021).

These results are arithmetically selective. Stover’s theorem, as quoted in the note, implies that only first-type arithmetic lattices can contain complex reflections for $PW=W/Z(W)$36, whereas second-type arithmetic lattices contain no complex reflections, even up to commensurability. The low-discriminant Picard groups are therefore a tractable testing ground inside a much more restrictive general landscape (Mark et al., 2021).

A higher-dimensional arithmetic example is the group constructed from the unique $PW=W/Z(W)$37-modular Hermitian $PW=W/Z(W)$38-module of signature $PW=W/Z(W)$39. Its projectivized reflection group $PW=W/Z(W)$40 is generated by $PW=W/Z(W)$41 complex reflections of order four. The mirrors of these reflections form a $PW=W/Z(W)$42-node Coxeter–Dynkin-type diagram $PW=W/Z(W)$43, indexed by sixteen points and sixteen affine hyperplanes in $PW=W/Z(W)$44; its automorphism group is

$PW=W/Z(W)$45

This group acts transitively on the $PW=W/Z(W)$46 mirrors and fixes a unique point $PW=W/Z(W)$47, and the $PW=W/Z(W)$48 mirrors are precisely those closest to $PW=W/Z(W)$49. A distinguished $PW=W/Z(W)$50-generator subsystem with diagram $PW=W/Z(W)$51 also generates the whole group (Basak, 2018).

6. Quotient geometry, abelian varieties, and derived categories

Complex projective reflection groups also appear through quotient constructions on abelian varieties. For an irreducible crystallographic reflection group scheme $PW=W/Z(W)$52 acting on an abelian variety $PW=W/Z(W)$53, the quotient $PW=W/Z(W)$54 is a weighted projective space, except for five explicitly enumerated quaternionic exceptions. If $PW=W/Z(W)$55 is a suitable $PW=W/Z(W)$56-linearized ample line bundle and the action is strongly crystallographic, then

$PW=W/Z(W)$57

is a polynomial algebra, and

$PW=W/Z(W)$58

This remains true in arbitrary finite characteristic, including characteristics dividing $PW=W/Z(W)$59. For the imprimitive monomial family one obtains

$PW=W/Z(W)$60

while numerous primitive Shephard–Todd cases have explicit weights tabulated case by case (Rains, 2023).

A categorical counterpart appears in the derived McKay correspondence for rank-two groups generated by order-two reflections. For

$PW=W/Z(W)$61

there is a semiorthogonal decomposition

$PW=W/Z(W)$62

where the $PW=W/Z(W)$63 are normalizations of irreducible components of the branch divisor and the $PW=W/Z(W)$64 are exceptional objects. The summands correspond exactly to conjugacy classes: the identity class gives $PW=W/Z(W)$65, conjugacy classes of reflections give the $PW=W/Z(W)$66, and all other non-identity, non-reflection classes give exceptional objects. Combined with Potter’s earlier treatment of $PW=W/Z(W)$67, this verifies the Orbifold Semiorthogonal Decomposition Conjecture for all finite $PW=W/Z(W)$68 generated by order-two reflections (Bhaduri et al., 2024).

Taken together, these developments show that complex projective reflection groups are not a single rigid class but a nexus of closely related constructions: scalar quotients of finite reflection groups, topological quotients detected by central braid-group reduction, arithmetic lattices in complex hyperbolic geometry, and algebro-geometric symmetry groups whose quotients retain explicit weighted-projective or categorical structures.