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Systems of imprimitivity for rank two quaternionic reflection groups

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Pith's one-line read This paper determines exactly which rank-two quaternionic reflection groups have more than one system of imprimitivity.

desk verdict A real repair-and-extend paper: Taylor corrects Cohen's imprimitive quaternionic reflection group tables, proves genuine conjugacies, and finds infinite systems of imprimitivity for some complex reflection groups; the main caveat is that completeness inherits unexamined parts of Cohen's structure theory. read the letter →

arxiv 2510.22134 v2 pith:RGKWSGHL submitted 2025-10-25 math.GR

Donald E Taylor This is my paper · ORCID
classification math.GR MSC 20D2520D60
keywords quaternionicreflectiongroupssystemsofimprimitivityranktwobinarypolyhedralconjugacyfinitequaternions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper revises the 1980 enumeration of imprimitive rank-two quaternionic reflection groups by determining which of them have more than one system of imprimitivity. The classification of such groups (Theorems 6.4–6.6) shows that extra systems are rare and correspond to conjugacies between groups that were previously listed as distinct. This leads to a corrected table of proper imprimitive groups, with some omitted groups added and several spurious duplicates identified. A striking consequence is that certain primitive complex reflection groups, when viewed as quaternionic reflection groups, admit infinitely many distinct systems of imprimitivity, even though they are primitive as complex groups.

What carries the argument

The central object is the standard imprimitive group G(K,H,φ), built from a finite subgroup K of the unit quaternions, a normal subgroup H, and an order-≤2 automorphism φ of K/H, acting on H^2 via diagonal matrices and the swap. A system of imprimitivity is a pair of orthogonal lines [[u,v]]. The paper shows that any additional system must have u of the form (1,1), (1,ai), (1,j), or (1,ck) in the binary dihedral case, or (1,rδ), (1,rj) in the polyhedral cases, and it uses explicit order-2 reflections R_{r,θ} = (1/√(1+r^2)) [[1,rθ],[−rθ,1]] both to test whether a candidate system is preserved and to conjugate one group to another. This conjugation machinery is what turns the classification of

What would settle it

Compute the normalizer of a standard group G(D_m,1,ψ_1) for m>2 and check whether any order-2 reflection outside the group preserves a second pair of lines; the paper's Theorem 6.4 says no such pair exists, so finding one would falsify it.

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Extended reading notes

Core claim

The central result is a complete classification of the rank-two imprimitive quaternionic reflection groups that possess more than one system of imprimitivity. For each such group, the possible alternative systems are explicitly listed: in the binary dihedral family (Theorem 6.4) extras occur exactly for certain small m and r; in the binary polyhedral families (Theorems 6.5 and 6.6) extras occur only for specific groups G(T,C2,ρ(δ)), G(T,1,ρ(δ)), G(O,1,ρ(δ)), G(I,1,ρ(j)), and the three groups with system [[(1,1),(1,−1)]]; and among extended binary polyhedral groups (Theorem 6.7) only C4⊡O, C4⊡2O, C4⊡I have extra systems. Conjugating by the explicit reflections R_{r,θ} realizes isomorphisms be

Load-bearing premise

The paper's enumeration rests on the 1980 structure theorem stating that every irreducible imprimitive quaternionic reflection group is conjugate to a standard group G(K,H,φ); if that theorem has further exceptions beyond the two lemmas corrected here, the list of systems of imprimitivity would be incomplete.

Editorial extensions

If this is right

  • The corrected Table 5 supersedes the earlier list; several groups previously thought distinct are now known conjugate, and missing entries are included.
  • The complex-type groups that in the classical tables are the primitive rank-two groups with 12, 18, and 30 reflections have infinitely many systems of imprimitivity as quaternionic reflection groups despite being primitive as complex reflection groups.
  • The monomial complex reflection groups of type (2m,m,2) for m>2 also have infinitely many quaternionic systems of imprimitivity.
  • The groups C4⊡O, C4⊡2O, and C4⊡I are imprimitive and conjugate to groups in the standard G(K,H,φ) tables, contrary to the earlier classification which had placed them among primitive groups.
  • The explicit reflection conjugators R_{r,θ} provide a constructive verification of every claimed conjugacy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the corrected enumeration is right, the symplectic-reflection and McKay-correspondence examples built from quaternionic reflection groups may need revisiting: the newly noted conjugacies could identify quotient singularities that were previously thought distinct.
  • The infinite families of systems of imprimitivity for complex-type groups show that imprimitivity is representation-dependent: the same abstract group can be simultaneously primitive as a complex reflection group and imprimitive as a quaternionic reflection group.
  • The method of using order-2 reflections as conjugators is a template for a rank n>2 analogue, though the paper does not address higher ranks; one could ask what extra systems appear for higher-rank imprimitive quaternionic reflection groups.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. The paper revisits Cohen's classification of imprimitive rank-two quaternionic reflection groups. Working inside Cohen's G(K,H,φ) framework, the author corrects omissions in the published tables (Remark 4.14), establishes new conjugacies among the standard copies (Theorems 7.1, 7.5, 7.7, 7.9), and proves the main structural results: Theorems 6.4–6.6 determine, for the binary dihedral and binary polyhedral families, all quaternionic reflection groups with more than one system of imprimitivity. Theorem 6.7 handles the extended binary polyhedral groups. Consequences include the fact that certain complex reflection groups—ST(12), ST(13), ST(22), and the imprimitive ST(2m,m,2)—have infinitely many systems of imprimitivity when considered as quaternionic groups (Remarks 7.4 and 7.6). Table 5 is the resulting revised list of proper imprimitive rank-two quaternionic reflection groups. The proofs are largely self-contained once Cohen's structure theorem is assumed, with explicit generators and direct matrix computations.

Significance. If correct, the paper fills known gaps in the imprimitive case of Cohen's classification and, more importantly, gives the first systematic determination of systems of imprimitivity in this setting. The discovery that primitive complex reflection groups of rank two can admit infinitely many quaternionic systems of imprimitivity is a notable phenomenon relevant to the McKay correspondence and symplectic resolutions. The manuscript is explicit: generators are given for each group, conjugating matrices are exhibited, and the key calculations are shown. The use of Magma is confined to exploration and validation, and the proofs do not depend on computer calculations. The main external input is Cohen's Theorem (2.2); this is a standard citation, and the paper's corrections to other statements in [7] do not, on inspection, invalidate the results.

minor comments (7)
  1. [Lemma 6.2 / Theorem 6.4] The phrase 'suppose that [[u,v]], [[e1,e2]] is a system of imprimitivity' is ambiguous: [[u,v]] and [[e1,e2]] are each systems, and the intended meaning is that both are systems (with [[e1,e2]] the standard one). Please rephrase, e.g., 'suppose that [[u,v]] is another system of imprimitivity in addition to the standard system [[e1,e2]].' Also, Corollary 6.3 has 'more then' for 'more than'.
  2. [Definition 4.16(1)] The standard copy of G(D_m,C_ℓ,ψ_r) is written with K=⟨ζ_m,j⟩. With the convention in §3 that D_m=⟨ζ_{2m},j⟩ (order 4m), the subscript should presumably be 2m to match Theorem 4.13 and the generators (4.3); if ζ_m is intentional, please explain the notational shift.
  3. [Theorem 6.4(iv)] The statement 'if and only if m=ℓ=1' for all c∈R should be read together with the proof: for G(D_1,1,ψ_1) all c are allowed, while in the ℓ=2 cases (m=1,2) the parameter c is restricted to ±1 (or 0). Please clarify in the statement to avoid confusion.
  4. [Lemma 4.17 proof] The proof contains the duplicated phrase 'generated generated by its elements of order two'. The argument that only Alt(4) fails to be generated by its elements of order two is also terse; a one-sentence justification would help.
  5. [Remark 7.4] The assertion that ST(12), ST(13), and ST(22) are 'the only primitive complex reflection groups of rank two all of whose reflections have order 2' is used to advertise the phenomenon. A reference or a brief justification would be helpful.
  6. [Sections 4 and 7] The completeness of Table 5 and of Theorems 6.4–6.6 relies on Cohen's Theorem (2.2) and on Lemma 4.5, which is quoted from [7, Lemma 2.4]. In light of the corrections to other parts of [7] (Remark 7.2 and §5), it would be helpful to add a sentence stating that these two results have been checked (e.g., with Magma) and are not affected by the corrections. This is a request for clarity, not a challenge to the mathematics.
  7. [References] Reference [13] lists the arXiv identifier without a year; add the year for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central theorems are derived by explicit matrix computation from Cohen's external G(K,H,φ) model.

full rationale

The main results (Theorems 6.4, 6.5, 6.6) are obtained by direct calculation: starting from the standard generators of G(D_m,C_ℓ,ψ_r), G(T,H,φ), G(O,H,φ), and G(I,H,φ), the paper solves for all possible vectors u such that [[u,v]] is a second system of imprimitivity. The restrictive Lemma 6.2 is proved from the actual reflection generators, not from the theorem it is used to prove. The conjugacy/isomorphism statements in Section 7 are supported by explicit conjugating matrices such as R_{r,θ} and T, and identifications with Shephard-Todd groups rely on the external classification in [10] plus the specific displayed generators after conjugation. The dependence on Cohen's Theorem (2.2) and Lemma 2.4 is an external structural assumption, not an input that already contains the paper's classification of systems of imprimitivity; the paper even corrects other parts of [7], showing it does not treat that source as infallible. Citations to the author's own book [10] are for standard facts about finite subgroups of S^3 and Shephard-Todd labels, which are independently checkable and are not used to define the target conclusions. No fitted parameter is renamed as a prediction, and no definition presupposes the claimed result. Thus there is no exhibited circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central derivation rests on standard finite-subgroup classification of S^3 and on Cohen's structure theorem for imprimitive quaternionic reflection groups. These are external theorems, not fitted parameters. No free parameters or invented entities are introduced.

assumptions (5)
  • standard math The finite subgroups of the multiplicative group of quaternions are exactly the cyclic groups and binary polyhedral groups D_m, T, O, I.
    Used throughout Section 4 to enumerate possible K; stated in Section 3 with reference to [10, 12].
  • domain assumption Every irreducible imprimitive quaternionic reflection group is conjugate to G(K,H,φ) with K a finite subgroup of S^3, H normal, and φ an automorphism of K/H of order ≤ 2 (Cohen [7, Theorem (2.2)]).
    The classification and Theorems 6.4–6.6 are built on this representation. It is cited from [7] and not reproved. The paper corrects Cohen's Lemma (2.3), so the remaining parts of [7] are load-bearing.
  • standard math The Shephard-Todd classification of finite complex reflection groups and the identification of ST groups by order and reflection counts (Table D.1 of [10]).
    Used in Theorems 7.3 and 4.7 to identify ST(12), ST(13), ST(22), and ST(2m,m,2).
  • domain assumption The classification of primitive rank-two complex reflection groups as C_d ◦_f K with K=T,O,I and d,f as in Cohen [5, §3].
    Used in Section 5 to construct extended binary polyhedral groups and in Theorem 5.3.
  • standard math For an irreducible imprimitive quaternionic group, the subspaces in a system of imprimitivity are pairwise orthogonal of dimension 1.
    Stated in Section 2 with reference to [10, Theorem 1.27]; used to normalize systems in Lemma 6.2.

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Pith. "Pith review of Systems of imprimitivity for rank two quaternionic reflection groups." pith.science (2026). https://pith.science/paper/RGKWSGHL

@misc{pith2026251022134,
  author       = {Pith},
  title        = {Pith review of: Systems of imprimitivity for rank two quaternionic reflection groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGKWSGHL}},
  note         = {Machine review of arXiv:2510.22134}
}
read the original abstract

We revise the enumeration of the imprimitive rank two quaternionic reflection groups, adding missing groups and establishing isomorphisms between groups in the published tables. The isomorphisms are obtained as a consequence of the determination of the reflection groups with more than one system of imprimitivity. We find that there are primitive complex reflection groups which have infinitely many systems of imprimitivity when represented as quaternionic reflection groups.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Namikawa--Weyl groups of symplectic quotient singularities

    math.SG 2026-07 accept novelty 6.5 of 10

    Every irreducible Weyl group arises as a factor of the Namikawa–Weyl group of some symplectic quotient singularity V/G with G a symplectic reflection group.

  2. Invariants in the cohomology of the complement of quaternionic reflection arrangements

    math.RT 2025-10 conditional novelty 6.0 of 10

    Invariant cohomology Poincaré polynomials of quaternionic reflection arrangements coincide with the complex cases except for imprimitive groups with non-cyclic K/H, where P(t^{1/3}) = 1+2t+...+2t^{n-2}+5t^{n-1}+4t^n.

Reference graph

Works this paper leans on

15 extracted references · 1 linked inside Pith · cited by 2 Pith papers

  1. [7]

    Arjeh M. Cohen. ‘Finite quaternionic reflection groups’.J. Algebra64(2) (1980), 293–324

    1980

  2. [1]

    ‘On the (non)existence of symplectic resolutions of linear quotients’.Math

    Gwyn Bellamy and Travis Schedler. ‘On the (non)existence of symplectic resolutions of linear quotients’.Math. Res. Lett.23(6) (2016), 1537–1564

    2016

  3. [2]

    ‘On parabolic subgroups of symplectic reflection groups’.Glasg

    Gwyn Bellamy, Johannes Schmitt, and Ulrich Thiel. ‘On parabolic subgroups of symplectic reflection groups’.Glasg. Math. J.65(2) (2023), 401–413

    2023

  4. [3]

    H. F. Blichfeldt.Finite Collineation Groups(University of Chicago Press, Chicago, 1917)

    1917

  5. [4]

    ‘The Magma algebra system

    Wieb Bosma, John Cannon, and Catherine Playoust. ‘The Magma algebra system. I. The user language’.J. Symbolic Comput.24(3-4) (1997), 235–265

    1997

  6. [5]

    Arjeh M. Cohen. ‘Finite complex reflection groups’.Ann. Sci. École Norm. Sup. (4)9(3) (1976), 379–436

    1976

  7. [6]

    Arjeh M. Cohen. ‘Finite quaternionic reflection groups’. Technical report, Technische Hogeschool Twente. Memorandum Nr. 229. 1978

    1978

  8. [8]

    Conway and Derek A

    John H. Conway and Derek A. Smith.On quaternions and octonions: their geometry, arithmetic, and symmetry(A K Peters, Ltd., Natick, MA, 2003)

    2003

Show all 15 references
  1. [9]

    W. C. Huffman and D. B. Wales. ‘Linear groups containing an involution with two eigenvalues −1’.J. Algebra45(2) (1977), 465–515

    1977

  2. [10]

    Lehrer and Donald E

    Gustav I. Lehrer and Donald E. Taylor.Unitary Reflection Groups,Australian Mathematical Society Lecture Series, V olume 20 (Cambridge University Press, Cambridge, 2009)

    2009

  3. [11]

    G. C. Shephard and J. A. Todd. ‘Finite unitary reflection groups’.Canad. J. Math.6(1954), 274–304

    1954

  4. [12]

    John V oight.Quaternion algebras,Graduate Texts in Mathematics, V olume 288 (Springer, Cham, 2021)

    2021

  5. [13]

    ‘An elementary classification of the quaternionic reflection groups of rank two’

    Shayne Waldron. ‘An elementary classification of the quaternionic reflection groups of rank two’. Technical report, arXiv:2509.01849. [math.GR]. 2025

    2025 arXiv

  6. [14]

    David B. Wales. ‘Linear groups of degreencontaining an involution with two eigenvalues−1. II’.J. Algebra53(1) (1978), 58–67

    1978

  7. [15]

    ‘On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C)’.Glasg

    Ryo Yamagishi. ‘On smoothness of minimal models of quotient singularities by finite subgroups of SLn(C)’.Glasg. Math. J.60(3) (2018), 603–634. D. E. Taylor, School of Mathematics and Statistics, The University of Sydney e-mail: Donald.Taylor@sydney.edu.au

    2018

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