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The lattice of normal reflection subgroups of an irreducible reflection group

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Pith's one-line read Normal reflection subgroups of a complex reflection group form a lattice indexed by the divisors of its reflection-orbit orders.

desk verdict Clean combinatorial reorganisation of normal reflection subgroups and the Shephard-Todd list around maximal collineation groups; solid for specialists, incomplete on quaternionic imprimitives. read the letter →

arxiv 2607.05466 v1 pith:WPUNJPZU submitted 2026-07-06 math.GR

Shayne Waldron This is my paper · ORCID
classification math.GR MSC 20F5520G2051F1520C25
keywords complexreflectiongroupsnormalsubgroupscollineationmaximalShephard-Toddclassificationorbitshiddenreflectionsquaternionic
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

Reflection groups are generated by reflections, so their reflection subgroups form a lattice ordered by the reflections they contain. This paper shows that the normal ones among those subgroups are completely determined by the conjugacy orbits of the rank-one reflection subgroups sitting on each root line. For a complex irreducible reflection group the resulting lattice is simply the lattice of divisors of the tuple of orbit orders, with meet and join given by coordinate-wise gcd and lcm. The same orbit data also produce natural generating sets. A second observation is that every complex reflection group sits as a collineation-preserving normal subgroup inside a unique maximal reflection group that shares its collineation group; the classical Shephard–Todd list can therefore be rewritten as a short list of these maximal groups together with their normal reflection subgroups. The paper works out the lattices, generators, quotients and collineation actions for both the primitive and imprimitive cases, and records how the picture changes for quaternionic groups.

What carries the argument

The reflection orbits Ra^G of the rank-one parabolic subgroups Ra (one per root line) together with the generation formula N=⟨∪ ̂Ra^G⟩ that produces every normal reflection subgroup from a choice of subgroups of the Ra; for complex groups this yields the unique divisor-label G(α1,…,αm).

What would settle it

Exhibit an irreducible complex reflection group possessing a normal reflection subgroup that cannot be written as the group generated by G-orbits of subgroups of its Ra’s, or that admits two distinct divisor labels.

Watch

Extended reading notes

Core claim

The lattice of normal reflection subgroups of an irreducible complex reflection group whose reflection orbits have orders (k1,…,km) is isomorphic to the divisor lattice of that tuple under coordinate-wise gcd and lcm; every complex reflection group is a collineation-preserving normal subgroup of a unique maximal reflection group sharing its collineation group.

Load-bearing premise

That every normal reflection subgroup is generated exactly by the G-conjugates of chosen subgroups of the rank-one reflection subgroups Ra, with uniqueness of labels guaranteed by the fact that every reflection is conjugate to a power of a generating reflection.

Editorial extensions

If this is right

  • The Shephard–Todd classification collapses to the short list of maximal reflection groups (G7, G11, G19 and the higher-rank primitives, plus the listed imprimitive families) together with their collineation-preserving normal subgroups.
  • The number of normal reflection subgroups of G(k1,…,km) is exactly the product of the numbers of divisors of the ki.
  • Quotients by normal reflection subgroups are again reflection groups and, except for four explicit families, are abelian of type C_{k1/α1} imes⋯ imes C_{km/αm}.
  • Collineation-preserving normal subgroups are precisely those without split orbits, equivalently those whose abelianisation has order equal to the product of the αi.
  • The same orbit data supply explicit minimal generating sets of reflections for every normal reflection subgroup.

Reading between the lines

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

  • The divisor-lattice description suggests that computer-algebra systems can enumerate normal reflection subgroups by pure arithmetic on the reflection type, without searching the full subgroup lattice.
  • The maximal-reflection-group viewpoint may streamline the construction of associated braid groups and Hecke algebras by fixing the collineation group first.
  • The appearance of multiple labels for quaternionic groups indicates that non-commutativity of the scalars can identify generators that remain independent over the complexes, offering a quantitative measure of how much more non-abelian the quaternionic theory is.
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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

2 major / 5 minor

Summary. The paper studies the lattice of normal reflection subgroups of a finite irreducible complex or quaternionic reflection group. For complex groups it shows that the conjugacy orbits of the rank-one parabolic reflection subgroups Ra completely determine the normal reflection subgroups via the generation formula N=⟨∪ ̂R_a^G⟩ (Lemma 2.1). When the reflection type is n1 C_{k1},…,nm C_{km}, the lattice is isomorphic to the divisor lattice of (k1,…,km) under coordinate-wise gcd/lcm (Theorem 2.1). It further proves that every complex reflection group is a collineation-preserving normal subgroup of a unique maximal reflection group sharing its collineation group (Lemma 3.1, Theorem 3.1), thereby reorganising the Shephard–Todd list as the maximal groups together with their collineation-preserving normal reflection subgroups. Explicit lattices, generators and quotients are computed for the primitive groups (especially G11, G7, G19) and the imprimitive families G(m,p,d); the same generation formula is applied to quaternionic groups, where labels need not be unique.

Significance. If the claims hold, the paper supplies a clean combinatorial description of the normal-reflection-subgroup lattice for every irreducible complex reflection group and a transparent reorganisation of the Shephard–Todd classification around maximal reflection groups. The abelianisation correspondence (Lemma 7.1, Corollary 7.1) and the explicit quotient calculations (Theorem 9.1) give concrete new information that can be used in invariant theory and representation theory. The Magma-assisted enumerations for the low-rank exceptional groups and the systematic treatment of generators via inherited reflections and dominos make the results immediately usable. The quaternionic discussion, while incomplete for the imprimitive families, correctly records the structural differences (non-unique labels, non-pointwise stabilisers) and therefore provides a useful starting point for further work.

major comments (2)
  1. Section 10 and the concluding remarks state that the normal reflection subgroups of the imprimitive quaternionic groups G(n,a,b,r) have not been fully determined. Lemma 2.1 still characterises them, but without an exhaustive list or a uniqueness criterion the claim that the method extends uniformly to the quaternionic case remains incomplete. Either complete the enumeration (or prove that labels become unique under additional hypotheses) or explicitly restrict the main theorems to the complex case and the primitive quaternionic groups already treated.
  2. Theorem 9.1 asserts that G/N is abelian except for four listed families. The verification relies on comparing group orders with the product of the α j (inequalities (9.70)) and on Magma checks for the exceptional primitive groups. A short independent argument that no further exceptions exist among the infinite imprimitive families would strengthen the claim; alternatively, the paper should state that the list is exhaustive only up to the Magma verification already performed.
minor comments (5)
  1. The date on the title page is July 8, 2026 and several arXiv identifiers in the references are likewise future-dated; these should be corrected or replaced by permanent identifiers before publication.
  2. Figures 1–13 are described only by text; if the journal permits, the actual lattice diagrams should be included so that the reader can verify the claimed inclusions and split orbits at a glance.
  3. The notion of a “domino” (Section 6) is introduced without a formal definition that covers the non-unique 3C2 case of G11; a one-sentence clarification would avoid ambiguity.
  4. In Example 4.1 the generators FZ, RF, ZR are introduced without an explicit matrix formula; a brief reference to the earlier generators of G11 would help the reader reconstruct them.
  5. Table 5 lists abelianisations of the primitive quaternionic groups; a short remark explaining why they are elementary 2-groups (or a reference) would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: lattice isomorphism and maximal-group presentation follow from conjugation-closed generation plus external facts (Cohen, Clifford, Shephard-Todd), not from self-definition or fitted inputs.

full rationale

The paper’s central claims (Lemma 2.1 / Theorem 2.1 lattice of normal reflection subgroups isomorphic to the coordinate-wise divisor lattice of the reflection type; Theorem 3.1 every complex reflection group is a collineation-preserving normal subgroup of a unique maximal reflection group) are derived directly from the definition of reflection subgroups Ra, the conjugation action, and two external classical facts: Cohen’s Lemma 4.11(iii) that every reflection is conjugate to a power of a generating reflection (used only for uniqueness of labels in the complex case) and Clifford’s theorem (used only for irreducibility of nontrivial normal subgroups of primitive groups). The generation formula N = ⟨∪ ĤR_a^G⟩ is not a re-definition of the target; it is the explicit characterisation of those subgroups that are closed under G-conjugation, which is precisely the definition of normality for reflection subgroups. Hidden reflections and the maximal reflection group are defined by spectral data and generation, then shown by direct matrix calculation (Lemma 3.1) to contain G as a normal subgroup of the same collineation group; no parameter is fitted and no uniqueness theorem is imported from the author’s prior work. Self-citations (Wal24–26, BW26) supply concrete generators and tables for examples and for the quaternionic extension; they are not invoked as black-box premises for the complex-case theorems. Quaternionic non-uniqueness of labels is explicitly recorded (Examples 10.1, 10.3) rather than hidden. Consequently the derivation chain is self-contained against the classical literature and contains no reduction of a claimed prediction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper works entirely inside the classical theory of finite unitary reflection groups. No free parameters are fitted; the only background axioms are standard group-theoretic and representation-theoretic facts together with the known classifications of complex and quaternionic reflection groups. Invented entities are purely definitional re-packagings of existing objects.

assumptions (4)
  • domain assumption Finite irreducible complex reflection groups are precisely the Shephard-Todd groups G(m,p,d) and G4–G37.
    Used throughout as the ambient list that is reorganised; cited from ST54 and LT09.
  • standard math Every reflection in a complex reflection group generated by a set R is conjugate to a power of an element of R (Cohen, Lemma 4.11(iii)).
    Invoked to guarantee uniqueness of labels (α1,…,αm) for normal reflection subgroups.
  • standard math Parabolic subgroups of complex (resp. quaternionic) reflection groups are themselves reflection groups (Steinberg; BST23).
    Used to identify the rank-one reflection subgroups Ra as the maximal rank-one parabolics.
  • domain assumption The quotient of a complex reflection group by a normal reflection subgroup is again a reflection group (BBR02).
    Background for the investigation of quotients in §9.
invented entities (2)
  • hidden reflection independent evidence
    purpose: A non-reflection matrix that becomes a reflection after scalar multiplication; used to define the maximal reflection group sharing a collineation group.
    Definitional packaging of an existing spectral phenomenon; no new physical or algebraic object is postulated.
  • domino (of a reflection orbit) independent evidence
    purpose: A combinatorial block of columns that partitions an orbit and generates no extra reflections; used to visualise conjugacy classes of reflection subgroups.
    Purely combinatorial re-description of existing conjugacy data; independent evidence is the Magma enumeration that confirms the partitions.

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Cite this review

Pith. "Pith review of The lattice of normal reflection subgroups of an irreducible reflection group." pith.science (2026). https://pith.science/paper/WPUNJPZU

@misc{pith2026260705466,
  author       = {Pith},
  title        = {Pith review of: The lattice of normal reflection subgroups of an irreducible reflection group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPUNJPZU}},
  note         = {Machine review of arXiv:2607.05466}
}
read the original abstract

The reflection subgroups of a reflection group have a natural lattice structure given by the reflections that they contain. By considering the conjugation action orbits of the reflection subgroups for a given root line, we are able to give an essentially combinatorial way to calculate the lattice of all the normal reflection subgroups of a given (finite irreducible) reflection group, and natural generators for them. Moreover, we observe that every complex reflection group is a normal subgroup of the unique maximal reflection group which shares its collineation group. Hence, we are able to present the Shephard-Todd classification of the complex reflection groups as a collection of maximal reflection groups, together with appropriate (collineation preserving) normal reflection subgroups. We investigate the quotients by the normal reflection subgroups, which are known to be reflection groups. We also consider the action of the collineation group on some appropriate small systems of lines, and how these results extend to quaternionic reflection groups. Some novel techniques are introduced, including the notion of a "hidden reflection", a combinatorial-geometric description of the reflection subgroups and the size of their conjugacy class, and the role played by the abelianisation of the reflection group.

Figures

Figures reproduced from arXiv: 2607.05466 by the authors.

Figure 1
Figure 1. The lattice of the 12 normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. The lattice of normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. The lattice of the maximal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The lattice of normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The lattice of normal reflection subgroups for [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: The inclusions between the primitive complex reflection groups [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: The normal reflection subgroup lattices for [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: The lattice of normal reflection subgroups for those [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: Some incidental inclusions, i.e., those not occurring as normal subgroups. [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: The abelianisations of the normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]
Figure 11
Figure 11. Figure 11: The lattice of normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]
Figure 12
Figure 12. Figure 12: A sublattice of normal reflection subgroups of [PITH_FULL_IMAGE:figures/full_fig_p038_12.png]
Figure 13
Figure 13. Figure 13: The quotient reflection groups G/N for the normal reflection subgroups N = Gα of G = G7 = G(2,3,3), arranged as in the lattice of [PITH_FULL_IMAGE:figures/full_fig_p051_13.png]

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Works this paper leans on

48 extracted references · 14 canonical work pages

  1. [1]

    On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

    On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups. arXiv e-prints , keywords =. doi:10.48550/arXiv.2604.00584 , archivePrefix =. 2604.00584 , primaryClass =

    work page Pith review arXiv doi:10.48550/arxiv.2604.00584

  2. [2]

    arXiv e-prints , keywords =

    An elementary classification of the quaternionic reflection groups of rank two. arXiv e-prints , keywords =. doi:10.48550/arXiv.2509.01849 , archivePrefix =. 2509.01849 , primaryClass =

    arXiv doi:10.48550/arxiv.2509.01849

  3. [3]

    The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$

    The geometry of the six quaternionic equiangular lines in H ^2. arXiv e-prints , keywords =. doi:10.48550/arXiv.2411.16766 , archivePrefix =. 2411.16766 , primaryClass =

    work page Pith review arXiv doi:10.48550/arxiv.2411.16766

  4. [4]

    Electron

    Buckley, Zachary and Waldron, Shayne , TITLE =. Electron. J. Linear Algebra , FJOURNAL =. 2026 , PAGES =. doi:10.13001/ela.2026.9905 , URL =

    work page doi:10.13001/ela.2026.9905 2026

  5. [5]

    Quotients et extensions de groupes de r\'

    Bessis, David and Bonnaf\'. Quotients et extensions de groupes de r\'. Math. Ann. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s002080100284 , URL =

    work page doi:10.1007/s002080100284 2002

  6. [6]

    Manuscripta Math

    Beck, Vincent , TITLE =. Manuscripta Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s00229-011-0438-9 , URL =

    work page doi:10.1007/s00229-011-0438-9 2011

  7. [7]

    Discrete complex reflection groups

    Discrete complex reflection groups. arXiv e-prints , keywords =. doi:10.48550/arXiv.2304.08941 , archivePrefix =. 2304.08941 , primaryClass =

    work page Pith review arXiv doi:10.48550/arxiv.2304.08941

  8. [8]

    Konishi, Yukiko and Minabe, Satoshi , TITLE =. J. Geom. Phys. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.geomphys.2025.105597 , URL =

    doi:10.1016/j.geomphys.2025.105597 2025

Show all 48 references
  1. [9]

    Matthew and Pfeiffer, G\"

    Douglass, J. Matthew and Pfeiffer, G\". On reflection subgroups of finite. Comm. Algebra , FJOURNAL =. 2013 , NUMBER =. doi:10.1080/00927872.2012.661005 , URL =

    2013 doi

  2. [10]

    Steinberg, Robert , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1964 , PAGES =. doi:10.2307/1994152 , URL =

    1964 doi

  3. [11]

    Bellamy, Gwyn and Schmitt, Johannes and Thiel, Ulrich , TITLE =. Glasg. Math. J. , FJOURNAL =. 2023 , NUMBER =. doi:10.1017/S0017089522000416 , URL =

    2023 doi

  4. [12]

    Shephard, G. C. and Todd, J. A. , TITLE =. Canad. J. Math. , FJOURNAL =. 1954 , PAGES =. doi:10.4153/cjm-1954-028-3 , URL =

    1954 doi

  5. [13]

    and Williams, Nathan F

    Arreche, Carlos E. and Williams, Nathan F. , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2023 , NUMBER =. doi:10.1017/S1474748021000323 , URL =

    2023 doi

  6. [14]

    and Williams, Nathan F

    Arreche, Carlos E. and Williams, Nathan F. , TITLE =. S\'. 2020 , PAGES =

    2020

  7. [15]

    Taylor, D. E. , TITLE =. J. Algebra , FJOURNAL =. 2012 , PAGES =. doi:10.1016/j.jalgebra.2012.04.033 , URL =

    2012 doi

  8. [16]

    and Taylor, Donald E

    Lehrer, Gustav I. and Taylor, Donald E. , TITLE =. 2009 , PAGES =

    2009

  9. [17]

    Finite collineation groups : with an introduction to the theory of operators and substitution groups

    Blichfeldt, Hans Frederik , address =. Finite collineation groups : with an introduction to the theory of operators and substitution groups. , year =

  10. [18]

    Introduction to complex reflection groups and their braid groups , SERIES =

    Brou\'. Introduction to complex reflection groups and their braid groups , SERIES =. 2010 , PAGES =. doi:10.1007/978-3-642-11175-4 , URL =

    2010 doi

  11. [19]

    Complex reflection groups, braid groups,

    Brou\'. Complex reflection groups, braid groups,. J. Reine Angew. Math. , FJOURNAL =. 1998 , PAGES =

    1998

  12. [20]

    , TITLE =

    Cohen, Arjeh M. , TITLE =. Ann. Sci. \'. 1976 , NUMBER =

    1976

  13. [21]

    , TITLE =

    Cohen, Arjeh M. , TITLE =. J. Algebra , FJOURNAL =. 1980 , NUMBER =. doi:10.1016/0021-8693(80)90148-9 , URL =

    1980 doi

  14. [22]

    arXiv e-prints , keywords =

    The quaternionic systems of imprimitivity for the reflection groups of rank two. arXiv e-prints , keywords =. doi:10.48550/arXiv.2601.17075 , archivePrefix =. 2601.17075 , primaryClass =

    doi

  15. [23]

    preprint , year = 2026, month = april, adsurl =

    The reflection orbits and normal reflection subgroups of the quaternionic reflection groups. preprint , year = 2026, month = april, adsurl =

    2026

  16. [24]

    Waldron, Shayne F. D. , TITLE =. 2018 , PAGES =. doi:10.1007/978-0-8176-4815-2 , URL =

    2018 doi

  17. [25]

    Arreche and Nathan F

    Carlos E. Arreche and Nathan F. Williams. Normal reflection subgroups. S\' e m. Lothar. Combin. , 84B:Art. 92, 12, 2020

    2020

  18. [26]

    Arreche and Nathan F

    Carlos E. Arreche and Nathan F. Williams. Normal reflection subgroups of complex reflection groups. J. Inst. Math. Jussieu , 22(2):879--917, 2023

    2023

  19. [27]

    Quotients et extensions de groupes de r\' e flexion

    David Bessis, C\' e dric Bonnaf\' e , and Rapha\" e l Rouquier. Quotients et extensions de groupes de r\' e flexion. Math. Ann. , 323(3):405--436, 2002

    2002

  20. [28]

    Abelianization of subgroups of reflection groups and their braid groups: an application to cohomology

    Vincent Beck. Abelianization of subgroups of reflection groups and their braid groups: an application to cohomology. Manuscripta Math. , 136(3-4):273--293, 2011

    2011

  21. [29]

    Finite collineation groups : with an introduction to the theory of operators and substitution groups

    Hans Frederik Blichfeldt. Finite collineation groups : with an introduction to the theory of operators and substitution groups. University of Chicago science series. University of Chicago Press, Chicago, 1917

    1917

  22. [30]

    Complex reflection groups, braid groups, H ecke algebras

    Michel Brou\' e , Gunter Malle, and Rapha\" e l Rouquier. Complex reflection groups, braid groups, H ecke algebras. J. Reine Angew. Math. , 500:127--190, 1998

    1998

  23. [31]

    Introduction to complex reflection groups and their braid groups , volume 1988 of Lecture Notes in Mathematics

    Michel Brou\' e . Introduction to complex reflection groups and their braid groups , volume 1988 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2010

    1988

  24. [32]

    On parabolic subgroups of symplectic reflection groups

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

    2023

  25. [33]

    Quaternionic MUB s in H^2 and their reflection symmetries

    Zachary Buckley and Shayne Waldron. Quaternionic MUB s in H^2 and their reflection symmetries. Electron. J. Linear Algebra , 42:1--20, 2026

    2026

  26. [34]

    Arjeh M. Cohen. Finite complex reflection groups. Ann. Sci. \' E cole Norm. Sup. (4) , 9(3):379--436, 1976

    1976

  27. [35]

    Arjeh M. Cohen. Finite quaternionic reflection groups. J. Algebra , 64(2):293--324, 1980

    1980

  28. [36]

    o tz Pfeiffer, and Gerhard R\

    J. Matthew Douglass, G\" o tz Pfeiffer, and Gerhard R\" o hrle. On reflection subgroups of finite C oxeter groups. Comm. Algebra , 41(7):2574--2592, 2013

    2013

  29. [37]

    Satake's good basic invariants for finite complex reflection groups

    Yukiko Konishi and Satoshi Minabe. Satake's good basic invariants for finite complex reflection groups. J. Geom. Phys. , 216:Paper No. 105597, 30, 2025

    2025

  30. [38]

    Lehrer and Donald E

    Gustav I. Lehrer and Donald E. Taylor. Unitary reflection groups , volume 20 of Australian Mathematical Society Lecture Series . Cambridge University Press, Cambridge, 2009

    2009

  31. [39]

    Vladimir L. Popov . Discrete complex reflection groups . arXiv e-prints , page arXiv:2304.08941, April 2023

    2023 arXiv

  32. [40]

    On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

    Gerhard Roehrle and Johannes Schmitt . On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups . arXiv e-prints , page arXiv:2604.00584, April 2026

    2026 arXiv

  33. [41]

    G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canad. J. Math. , 6:274--304, 1954

    1954

  34. [42]

    Differential equations invariant under finite reflection groups

    Robert Steinberg. Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. , 112:392--400, 1964

    1964

  35. [43]

    D. E. Taylor. Reflection subgroups of finite complex reflection groups. J. Algebra , 366:218--234, 2012

    2012

  36. [44]

    Shayne F. D. Waldron. An introduction to finite tight frames . Applied and Numerical Harmonic Analysis. Birkh\" a user/Springer, New York, 2018

    2018

  37. [45]

    The geometry of the six quaternionic equiangular lines in H ^2

    Shayne Waldron . The geometry of the six quaternionic equiangular lines in H ^2 . arXiv e-prints , page arXiv:2411.16766, November 2024

    2024 arXiv

  38. [46]

    An elementary classification of the quaternionic reflection groups of rank two

    Shayne Waldron . An elementary classification of the quaternionic reflection groups of rank two . arXiv e-prints , page arXiv:2509.01849, September 2025

    2025 arXiv

  39. [47]

    The quaternionic systems of imprimitivity for the reflection groups of rank two

    Shayne Waldron . The quaternionic systems of imprimitivity for the reflection groups of rank two . arXiv e-prints , page arXiv:2601.17075, January 2026

    2026

  40. [48]

    The reflection orbits and normal reflection subgroups of the quaternionic reflection groups

    Shayne Waldron . The reflection orbits and normal reflection subgroups of the quaternionic reflection groups . preprint , 2026

    2026

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