Complex Reflection Groups

• Complex Reflection Groups are finite subgroups of GL(V) generated by reflections that fix hyperplanes in an n-dimensional complex space.
• They are classified by Shephard–Todd into infinite families and exceptional cases, with structures that reveal intricate combinatorial and invariant properties.
• These groups underpin various applications in algebraic geometry, braid group theory, and mathematical physics through their connections to invariant theory and combinatorial models.

A complex reflection group is a finite subgroup $G \subset GL(V)$ of an $n$-dimensional complex vector space $V$, generated by elements (reflections) each fixing a hyperplane pointwise and acting with a single nontrivial eigenvalue. These groups generalize real (Coxeter) reflection groups and exhibit rich algebraic, combinatorial, geometric, and topological structures. They were classified by Shephard and Todd into infinite families and exceptional cases, initiating deep connections to invariant theory, braid groups, algebraic combinatorics, and mathematical physics.

1. Classification and Structure

The classification of irreducible complex reflection groups is due to Shephard–Todd. Every such group is either an imprimitive group of the form $G(m,p,n)$ (where $m, n \geq 1$, $p \mid m$), or one of 34 exceptional groups $G_4, \dots, G_{37}$ (Etingof et al., 2010). The group $G(m,p,n)$ consists of $n \times n$ monomial matrices with entries mth roots of unity and determinant an $(m/p)$th root of unity. Special cases recover real finite reflection groups: $G(1,1,n) \simeq S_n$, $G(2,1,n)$ is the hyperoctahedral group ($B_n$), and $G(2,2,n)$ is the even-signed group ($D_n$) (Burnham-Schmidt et al., 4 Oct 2025).

Complex reflection groups admit a matrix or wreath-product realization, and their structure supports two distinct classes of reflections:

• Transposition-like: Order two, e.g., $[(i\ j);\, a]$ swaps $i$, $j$ and multiplies by roots of unity.
• Diagonal: Scalar diagonal matrices fixing a hyperplane, with order dividing $m$ (Lewis et al., 2021).

2. Invariant and Coinvariant Theory

For any complex reflection group $W$, Chevalley–Shephard–Todd proved that the invariant algebra $S^W := \mathbb{C}[V]^W$ is a polynomial algebra generated by $n$ homogeneous invariants of degrees $d_1 \leq \dots \leq d_n$ (Reiner et al., 2016, Briggs, 2017). The coinvariant algebra $A = S/I$ with $I = S\cdot S^W_+ = (f_1,\dots,f_n)$ is a complete intersection of codimension $n$. The spectrum of $(d_i)$ and codimension data underpins much of the algebraic and combinatorial structure (Briggs, 2017).

Well-generated ("duality") groups—those generated by $n$ reflections—are characterized by a degree-codegree duality: $d_i+d_i^* = d_n$ for all $i$, where $d_i^*$ are the codegrees. These include $G(m,1,n)$ for all $m,n$, $G(m,m,2)$, and certain exceptional $G_k$ (Briggs, 2017).

3. Reflection Length, Orders, and Posets

Define the reflection length $\ell(g)$ of $g \in G$ as the minimal number of reflections whose product is $g$, and codimension $\operatorname{codim}(g) = n - \dim V^g$. While $\operatorname{codim}(g) \leq \ell(g)$ always, equality $\ell(g) = \operatorname{codim}(g)$ throughout $G$ only occurs for Coxeter groups and $G(m,1,n)$ (Foster-Greenwood, 2012). In $G(m,p,n)$ (with $p > 1$ or in exceptional types), there exist elements with $\ell > \operatorname{codim}$, due to $p$-connected diagonal elements which are nonreflection atoms of the codimension poset (Foster-Greenwood, 2012). This discrepancy controls the necessity of higher-degree generators in the Hochschild cohomology of skew group algebras.

Both reflection length and codimension induce partial orders on $G$, studied using character-theoretic and algebraic techniques (Foster-Greenwood, 2012).

4. Hurwitz Action, Reflection Factorizations, and Quasi-Coxeter Elements

Reflection factorizations and their Hurwitz orbits encode deep combinatorial and group-theoretic information. For $g \in G(m,p,n)$, the Hurwitz action of the braid group $B_k$ on $k$-tuples $(t_1,\ldots,t_k)$ of reflections (preserving the product) relates all shortest reflection factorizations. The number of Hurwitz orbits and transitivity criteria are given via explicit formulas involving invariant-theoretic data (cycle partitions, gcd of weights) (Lewis et al., 2021).

An element $g$ is quasi-Coxeter if it has a minimal-length reflection factorization generating $G$; such elements are precisely characterized by conditions on cycle-weights and their generation of $\mathbb{Z}/m\mathbb{Z}$ and $p\,(\mathbb{Z}/m\mathbb{Z})$ (Lewis et al., 2021). The Hurwitz action is transitive on shortest factorizations of quasi-Coxeter elements.

5. Arrangements, Braid Groups, and Crystallographic Extensions

The arrangement $\mathcal{A}$ of reflecting hyperplanes under $G$ gives rise to a complex hyperplane complement $X = V \setminus \bigcup_{H \in \mathcal{A}} H$, fundamental group $P_W$ ("pure braid group"), and orbit complement $B_W = \pi_1(X/G)$ ("generalized braid group"). Short exact sequences encode the relationship between these groups and $W$ (Marin, 2015).

Quotients $B_W/[P_W,P_W]$ yield crystallographic groups acting on tori of rank equal to the number of hyperplanes, with holonomy group $W/Z(W)$. For $2$-subgroups, the corresponding Bieberbach groups are torsion-free, giving rise to flat manifolds with holonomy relating directly to the group-theoretic structure of $W$ (Marin, 2015).

Crystallographic complex reflection groups $W = G \ltimes \Lambda$ (with $G$ finite and $\Lambda$ a $G$-invariant full rank lattice) generalize affine Weyl groups. Steinberg's theorem for regular orbits being off all mirrors typically holds, with exceptions isolated to a handful of low-dimensional cases (Puente et al., 2018).

6. Combinatorics and Homological Aspects

Complex reflection groups underpin geometric and combinatorial objects such as noncrossing partitions, parking functions, and pinnacle sets. In the well-generated setting, the numbers of noncrossing partitions and parking functions are given by rational Catalan numbers and generalizations: $$|NC(W)| = \prod_{i=1}^n \frac{h + d_i}{d_i},\quad |\mathrm{Park}^{NC}(W)| = (h+1)^n,$$ where $h=d_n$ is the Coxeter number. Algebraic and combinatorial models of parking spaces are equivariantly isomorphic, and these correspondences extend to Fuss-Catalan cases (Miller, 2023, Stack, 4 Feb 2025).

Invariant differential derivations, homology in the order complexes arising from reflection arrangements, and the representation-theoretic realization of top homology as ribbon modules or Specht modules are universal phenomena in the well-generated setting (Miller, 2011, Reiner et al., 2016, Briggs, 2017).

7. Applications and Further Directions

Complex reflection groups have profound consequences in algebraic geometry (orbit spaces, cohomology rings, Kähler flat manifolds), mathematical physics (e.g., elliptic Calogero–Moser Hamiltonians, Dunkl operators), combinatorics (noncrossing objects, parking functions), invariant theory, and representation theory (Etingof et al., 2010, Briggs, 2017, Stack, 4 Feb 2025). The paper of normal reflection subgroups, braid group actions, and their cohomological invariants remains active (Arreche et al., 2020). The precise topological, geometric, and representation-theoretic properties of the exceptional groups and their generalizations continue to drive research at the interface of combinatorics, geometry, and algebra.

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References

• (Etingof et al., 2010) On elliptic Calogero-Moser systems for complex crystallographic reflection groups
• (Miller, 2011) Reflection arrangements and ribbon representations
• (Foster-Greenwood, 2012) Comparing Codimension and Absolute Length in Complex Reflection Groups
• (Marin, 2015) Crystallographic groups and flat manifolds from complex reflection groups
• (Reiner et al., 2016) Invariant derivations and differential forms for reflection groups
• (Briggs, 2017) Matrix Factorisations Arising From Well-Generated Complex Reflection Groups
• (Puente et al., 2018) Steinberg's theorem for crystallographic complex reflection groups
• (Arreche et al., 2020) Normal Reflection Subgroups of Complex Reflection Groups
• (Lewis et al., 2021) The Hurwitz action in complex reflection groups
• (Miller, 2023) Rational Catalan Numbers for Complex Reflection Groups
• (Haladjian, 28 Apr 2024) A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations
• (Stack, 4 Feb 2025) Parking Spaces for Complex Reflection Groups
• (Burnham-Schmidt et al., 4 Oct 2025) Pinnacles for Complex Reflection Groups
• (Tachikawa et al., 2019) Reflection groups and 3d $\mathcal{N}\ge6$ SCFTs