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arxiv: 2406.17509 · v3 · submitted 2024-06-25 · 🧮 math-ph · cond-mat.other· math.MP

Affine subgroups of the affine Coxeter group with the same Coxeter number

Nazife Ozdes Koca , Mehmet Koca This is my paper

Pith reviewed 2026-05-24 00:22 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.othermath.MP
keywords Coxeter groupsaffine subgroupsgraph foldingroot systemsCoxeter numberquasicrystallographydihedral subgroups
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The pith

Graph folding on Coxeter diagrams generates affine subgroups that share the Coxeter number of the original groups.

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

The paper constructs affine subgroups of the affine Coxeter groups W(An), W(Dn), and W(En) that possess the same Coxeter number as their parent groups by applying a graph folding technique to the corresponding Coxeter diagrams. Specific mappings include deriving the affine groups W(Cn) and W(Bn) from W(A2n-1) and W(D2n-1), respectively, along with W(F4), W(H3), and W(H4) obtained from W(E6), W(D6), and W(E8). It also presents a general construction for affine dihedral subgroups and notes their relevance to planar quasicrystallography, while employing sets of orthonormal vectors for most root systems and a special non-orthogonal set for W(An) to aid lattice and cell constructions.

Core claim

Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding. A general construction of the affine dihedral subgroups is introduced, some of which describe the existing planar quasicrystallography.

What carries the argument

The graph folding technique on Coxeter diagrams, which maps the diagram of a parent affine Coxeter group to a diagram of a subgroup while keeping the Coxeter number unchanged.

Load-bearing premise

Applying the graph folding operation to the Coxeter diagram always yields a subgroup whose Coxeter number equals that of the parent group.

What would settle it

Compute the Coxeter number after folding the diagram of W(E6) to obtain W(F4) and verify whether it equals 12, the Coxeter number of W(E6).

read the original abstract

Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding. The latter two are the non-crystallographic groups where W(H3) plays a special role in the quasicrystallographic structures with icosahedral symmetry. A general construction of the affine dihedral subgroups is introduced, some of which, describe the existing planar quasicrystallography. In the construction of the root systems, sets of orthonormal vectors are used but a special non-orthogonal set of vectors in the formulation of the root system of W(An) is also introduced which has practical applications in the construction of the lattices An and An* and their Delone and Voronoi cells.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to construct, via graph folding of Coxeter diagrams, affine subgroups of the affine Coxeter groups W(A_n), W(D_n) and W(E_n) that share the same Coxeter number as the parent group. It states that W(C_n) and W(B_n) arise from folding W(A_{2n-1}) and W(D_{2n-1}), respectively, and that folding the affine groups W(E6), W(D6) and W(E8) produces the groups W(F4), W(H3) and W(H4). A general construction of affine dihedral subgroups is introduced, with some describing planar quasicrystallography, and root-system constructions are discussed using orthonormal vectors together with a special non-orthogonal set for type A_n.

Significance. If the constructions are valid and the terminology is corrected, the work would supply a uniform graph-folding method for producing Coxeter subgroups that preserve the Coxeter number, with potential relevance to quasicrystal symmetries (especially icosahedral cases involving W(H3)) and to explicit lattice constructions for A_n and A_n^*.

major comments (1)
  1. [Abstract] Abstract: the sentence 'The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding' is factually incorrect. W(F4) is a finite crystallographic Coxeter group and W(H3), W(H4) are finite non-crystallographic groups; none are affine Coxeter systems. This directly contradicts the title and the central claim that the constructions produce affine subgroups of affine Coxeter groups, undermining the asserted uniformity of the folding technique for preserving both affinity and Coxeter number.
minor comments (1)
  1. [Abstract] The abstract correctly notes that 'the latter two are the non-crystallographic groups' but the preceding clause incorrectly applies the label 'affine groups' to all three examples; this internal inconsistency should be removed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying the terminological error in the abstract. We agree that the description of the groups obtained from folding the E-series affine Coxeter groups requires correction, as W(F4), W(H3) and W(H4) are finite rather than affine. We will revise the manuscript to clarify the distinction while preserving the core constructions and the uniform folding method for preserving Coxeter numbers.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the sentence 'The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding' is factually incorrect. W(F4) is a finite crystallographic Coxeter group and W(H3), W(H4) are finite non-crystallographic groups; none are affine Coxeter systems. This directly contradicts the title and the central claim that the constructions produce affine subgroups of affine Coxeter groups, undermining the asserted uniformity of the folding technique for preserving both affinity and Coxeter number.

    Authors: We acknowledge the factual inaccuracy. The cited sentence erroneously labels W(F4), W(H3) and W(H4) as affine groups; these are finite Coxeter groups (crystallographic for F4, non-crystallographic for H3 and H4). The graph-folding constructions applied to the affine groups W(E6), W(D6) and W(E8) produce subgroups isomorphic to these finite groups that share the same Coxeter number. In contrast, the A_n and D_n foldings correctly yield the affine groups W(C_n) and W(B_n). We will revise the abstract, introduction and relevant sections to state explicitly that the E-series cases produce finite subgroups via folding while the A/D cases produce affine subgroups, all preserving the Coxeter number. The uniformity of the technique lies in the preservation of the Coxeter number under folding, not in every resulting group being affine. We will also adjust the title if necessary to read 'Subgroups with the same Coxeter number obtained by graph folding from affine Coxeter groups' or similar to avoid implying all subgroups are affine. These changes will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions apply standard graph folding without self-referential reduction

full rationale

The paper constructs affine subgroups of W(An), W(Dn), W(En) via graph folding and states that the resulting groups (including W(Cn), W(Bn) from folding W(A2n-1), W(D2n-1)) have the same Coxeter number as the parent. This equality is presented as following from the folding operation on the diagrams rather than being assumed by definition or fitted to match. No equations, fitted parameters, or predictions appear that reduce to inputs by construction. The graph-folding technique is invoked as an existing method; any prior citations for it are not load-bearing for the central claim, as the application here produces explicit new diagrams whose properties can be checked independently in Coxeter theory. The noted mislabeling of finite groups as affine does not create a circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the standard axioms of Coxeter groups and root systems; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Graph folding on a Coxeter diagram yields a subgroup whose Coxeter number equals that of the parent group.
    Invoked by the statement that the constructed subgroups have the same Coxeter number.
  • standard math Standard reflection representation and root-system axioms for affine Coxeter groups.
    Background structure assumed for all constructions.

pith-pipeline@v0.9.0 · 5737 in / 1367 out tokens · 19083 ms · 2026-05-24T00:22:05.900133+00:00 · methodology

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Reference graph

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