Quaternionic Reflection Groups

Updated 23 October 2025

Quaternionic reflection groups are finite subgroups of GL(V) generated by automorphisms that fix quaternionic hyperplanes, extending classical real and complex cases with noncommutative properties.
Their construction leverages Clifford algebra and spinorial induction to compute root systems and classify group structures, enhancing our understanding of symmetry and representation theory.
Applications span lattice geometry, gauge theory moduli spaces, and quantum symmetry, making these groups pivotal in linking invariant theory with physical and geometric phenomena.

Quaternionic reflection groups are finite subgroups of the general linear group GL(V), where V is a quaternionic vector space, generated by elements called quaternionic reflections—automorphisms that fix a quaternionic hyperplane pointwise. These groups play central roles in geometry, invariant theory, representation theory, and mathematical physics. Their structure generalizes both real and complex reflection groups but exhibits new phenomena arising from both the noncommutativity of quaternions and their richer algebraic structure. Below, key aspects are organized to facilitate deep understanding.

1. Group Structure and Symplectic Representations

Quaternionic reflection groups, such as G = (Q₈ × D₈)/ℤ₂ ⊆ Sp₄(ℂ), exemplify hybrid symmetry. Here, Q₈ is the quaternion group ⎧±1, ±I, ±J, ±K⎫, realized via 2×2 matrices (e.g., I = diag(i, –i)), and D₈ is the dihedral group with standard 2-dimensional representation. The group G is the quotient of their direct product by identifying nontrivial central elements (–1) and acts on ℂ⁴ ≅ ℂ² ⊠ ℂ², where "⊠" denotes tensor product. The Q₈ action preserves a symplectic form on ℂ², while D₈ preserves an orthogonal form; their tensor product induces a symplectic structure, embedding G in Sp₄(ℂ). G occurs naturally as a subgroup of the wreath product Q₈² ⋊ S₂. Symplectic reflections in G correspond to noncentral involutions and fall into five explicit conjugacy classes, each represented by an element such as {±(I, ρ)} or {±(Id, σ)}, dictating the Poisson geometry of the quotient (Bellamy et al., 2011).

2. Clifford Algebra, Spinorial Induction, and Quaternionic Representations

Clifford algebra presents a systematic mechanism for describing reflection groups, streamlining the geometric interpretation of quaternionic roots and representations (Dechant, 2012, Dechant, 2016). Reflections in a hyperplane normal α are encoded by sandwiching: v' = –α v α, while general orthogonal transformations act via versors: v' = ±Ā v A (where Ā is the reverse of multivector A). In three dimensions, the even Clifford algebra is isomorphic to quaternions ℍ; root systems of rank‑3 Coxeter groups such as A₃, B₃, H₃ relate directly to sets of pure quaternions. Two reflections generate a rotation: a'' = m n a n m = R a Ŕ, with the rotor R = m n, exposing the spinorial structure behind quaternionic representation theory.

Spinorial induction allows construction of 4D exceptional root systems directly from 3D ones. Explicitly, A₃ → D₄, B₃ → F₄, H₃ → H₄; the 4D root systems realized as sets of spinors from 3D reflections encode phenomena such as triality and automorphism groups of the Platonic solids. This induction provides uniform understanding for structures that, in traditional approaches, require high-dimensional ambient spaces (e.g., folding E₈ to H₄). Binary polyhedral groups associated with quaternionic representations are efficiently computed as sets of Clifford spinors (Dechant, 2012, Dechant, 2016).

3. Classification via Reflection Systems

Recent advances provide an elementary classification of rank‑two quaternionic reflection groups using the notion of a "reflection system"—defined as a subset L of a finite quaternionic subgroup K (such as Q₈, cyclic, dicyclic, or binary polyhedral groups) satisfying identity inclusion, generation, and closure under a binary operation a ∘ b := a b⁻¹ a. Every group G = G(K, L, H) is canonically presented by diagonal reflections (from H ⊆ K) and nondiagonal ("exchange type") reflections (from L), with group order |G| = 2|H||K| and number of reflections 2|H| + |L| – 2. In the dicyclic case, reflection systems L_{(a,b)}ⁿ and suitable choices of normal subgroup H produce infinite families and new, previously missing, groups (e.g., four nonisomorphic groups of order 192 with 22 reflections) (Waldron, 2 Sep 2025). This systematic framework unifies complex and quaternionic cases, correcting earlier incomplete classifications.

4. Invariant Theory and Quotients

The invariant theory of quaternionic reflection groups extends Chevalley–Shephard–Todd type theorems to supersingular abelian varieties with quaternionic endomorphism algebra (Rains, 2023). In the generic case, the quotient of the abelian variety by a quaternionic reflection group action yields a weighted projective space, as the invariant section ring is polynomially generated. However, five explicit exceptional cases exhibit failure of this polynomial structure, revealing deeper subtleties (non-Cohen–Macaulay rings, extra syzygies) when the group or the equivariant line bundle structure does not align with "strongly crystallographic" hypotheses. Such exceptions mark boundaries of analogue results to classical settings in finite characteristic.

For linear quotient singularities, Cartaya and Griffeth generalize Haiman's coinvariant conjecture: for a quaternionic reflection group W acting on V, the "zero fiber" ring of π : V → V/W admits a (g+1)ⁿ-dimensional quotient, where g = 2N/n (N = number of reflections, n = quaternionic dimension). This is achieved via representation theory of symplectic reflection algebras, whose irreducible modules provide sharp lower bounds for the dimension of the zero‑fiber ring (Cartaya et al., 31 Jan 2024).

5. Cohomology, Arrangements, and Topology

The paper of the hyperplane arrangement complement associated with a quaternionic reflection group yields a recursive computation for the Poincaré polynomial pₐ(t) of the cohomology: pₐ(t) = p₍A⁰₎(t) + t·p₍A₀₎(t). The cohomology appears only in degrees divisible by three, reflecting a fundamental shift from the complex case (Griffeth et al., 21 Oct 2025). The Orlik–Solomon algebra, defined on generators corresponding to hyperplanes with grading shifts, models the cohomology algebra, and its graded ranks relate to the Möbius function of the intersection lattice. In almost all irreducible cases, pₐ(t) and related codimension polynomials factor into irreducible factors with positive integer coefficients, nearly always linear, except for precisely three exceptional groups where a quadratic factor arises, signalling deeper combinatorial or representation-theoretic phenomena.

6. Explicit Constructions, Geometry, and Applications

Quaternionic reflection group theory underpins constructions across geometry, algebra, and physics:

Lattice Geometry and Symmetry: Quaternionic representations encode cubic and pyritohedral lattice symmetries, delivering explicit lattice vector formulas and reflection plane identification linked to Coxeter diagrams (e.g., fcc/bcc/sc lattices and embedding polyhedra such as the pseudoicosahedron) (Koca et al., 2015).
Symmetry of Quantum Structures: Primitive quaternionic reflection groups yield the maximal set of five mutually unbiased bases in H², fixing projective line systems and producing spherical 3‑designs with optimal bounds (Buckley et al., 2 Sep 2025). Their action realizes sets such as the six equiangular lines in H² (orbit of 2·A₆), and more extensive line packings and designs (Waldron, 25 Nov 2024).
Moduli Spaces in Gauge Theory: For 3d N=5 superconformal field theories, the moduli space takes the form ℍ^{2r}/Γ, where Γ is a discrete quaternionic reflection group. The Hilbert series extracted from superconformal indices matches the Molien formula for the invariants, confirming explicit ring structures and linking discrete group quotient data to gauge theory vacua (Deb et al., 6 Mar 2024).
Arithmetic and Hyperbolic Geometry: Quaternionic (and complex) reflection groups generate arithmetic lattices such as Picard modular groups and quaternionic hyperbolic lattices, with explicit matrix representations and index calculations mediating transmutation between reflection groups and arithmetic constructions (Mark et al., 2021).

7. Mathematical Formulations and Notational Conventions

Key algebraic constructions include:

Symplectic Reflection Algebra:

$H_c(G) = T(V) \rtimes G / \langle vw - wv - \sum_{s \in S} c(s) \omega_s(v, w) s \rangle$

where $\omega_s$ is the projected symplectic form onto the reflection direction, and $S$ enumerates the symplectic reflections.

Canonical Group Presentation (Rank Two):

$G(K,L,H) = \left\{ \begin{pmatrix} b & 0 \ 0 & b_\gamma h \end{pmatrix} : b \in K, h \in H \right\} \cup \left\{ \begin{pmatrix} 0 & b \ b^{-1} & 0 \end{pmatrix} : b \in L \right\}$

with closure property $a \circ b = a b^{-1} a$ for $a, b \in L$ .

Reflection Action on Sections:

$r^* s = \chi(r) s, \qquad \chi(r)^{o(r)} = 1$

where $\chi(r)$ is a root of unity and $o(r)$ is the order.

Zero Fiber Coinvariants:

$g = \frac{2N}{n}, \qquad \text{Quotient ring dimension} = (g+1)^n$

These formulas articulate the invariant, cohomological, and quotient ring structures central to applications and theoretical development.

In summary, quaternionic reflection groups embody a central and sophisticated theme in modern algebraic, geometric, and physical mathematics. Their classification, representation theory, invariant rings, cohomology, and applications constitute a rich tapestry deepening the understanding of symmetry, singularity resolution, and group actions beyond the reach of classical reflection group theory. The interplay of Clifford algebra, symplectic quotient constructions, and arithmetic modular groups continues to reveal new phenomena and unify diverse domains under the rubric of quaternionic symmetry.