pith. sign in

arxiv: 2606.26082 · v2 · pith:Z7ITH46Onew · submitted 2026-06-24 · 🧮 math.GR

Automorphisms of the Artin group of type D₅

Luis Paris , Ignat Soroko This is my paper

Pith reviewed 2026-06-26 05:37 UTC · model grok-4.3

classification 🧮 math.GR
keywords Artin groupsautomorphismsspherical typetype D5center quotientgroup automorphismsCoxeter groups
0
0 comments X

The pith

The automorphism group of the Artin group of type D5 and of its quotient by the center have been determined.

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

The paper finishes the classification of automorphisms for all Artin groups of spherical type Dn by treating the single remaining case n=5. It computes both the full automorphism group of the D5 Artin group and the automorphism group of the quotient obtained by dividing out the center. A reader would care because the result closes a case-by-case program that had already covered every other spherical Dn group. The work shows that the D5 case requires no new exceptions and follows the pattern established for larger and smaller n.

Core claim

For the Artin group of type D5 the authors determine its automorphism group and the automorphism group of its quotient by the center. This settles the only remaining case n=5 in the classification of automorphisms of Artin groups of spherical type Dn.

What carries the argument

The Artin group of type D5, presented by the standard Coxeter diagram of type D5 together with its center and the quotient by that center.

If this is right

  • The full list of automorphisms is now known for every spherical type Dn Artin group.
  • The same determination applies to the quotient by the center in the D5 case.
  • No additional generators or relations are needed to describe the automorphism groups beyond those used for other Dn.
  • The outer automorphism group of the D5 Artin group is now explicitly known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that any future computation of Out groups for related Artin groups can safely reuse the D5 case as a template.
  • Explicit matrix or permutation representations of the automorphisms could now be written down for D5 and checked against known presentations.
  • The same quotient-by-center construction might be examined for non-spherical Artin groups to test whether the pattern persists.

Load-bearing premise

The techniques and results already established for spherical type Dn Artin groups when n is not 5 extend without obstruction to the n=5 case.

What would settle it

An automorphism of the D5 Artin group (or of its center quotient) whose action on the standard generators lies outside the families described in the determination would show the classification is incomplete.

Figures

Figures reproduced from arXiv: 2606.26082 by Ignat Soroko, Luis Paris.

Figure 1
Figure 1. Figure 1: The Coxeter graphs of types D5 and A4 . Theorem 1 Let A = A[D5] and C2 = Z/2Z. Then Aut(A) = Inn(A) ⋊ ⟨χ⟩ ≃ A/Z(A) ⋊ C2. In particular, Out(A) ≃ C2 . Theorem 2 Let A = A[D5], A = A/Z(A), and χ be the automorphism induced by χ. Then Aut(A) = Inn(A) ⋊ ⟨χ⟩ ≃ A/Z(A) ⋊ C2. In particular, Out(A) ≃ C2 . Thus the descriptions of Aut(A[D5]) and of Aut(A[D5]/Z(A[D5])) have the same form as the correspond￾ing descrip… view at source ↗
read the original abstract

For the Artin group of type $D_5$, we determine its automorphism group and the automorphism group of its quotient by the center. This settles the only remaining case, $n=5$, in the classification of automorphisms of Artin groups of spherical type $D_n$.

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

0 major / 2 minor

Summary. The manuscript determines the automorphism group of the Artin group of type D_5 and the automorphism group of its quotient by the center. This completes the classification of automorphisms for all spherical type D_n Artin groups by resolving the remaining case n=5.

Significance. If correct, the result completes a classification begun in prior work on D_n for n≠5, providing a full determination of Aut(A) and Aut(A/Z(A)) for this infinite family of Artin groups. The contribution lies in extending established techniques to the exceptional case without introducing new phenomena that obstruct the classification.

minor comments (2)
  1. The introduction should include an explicit statement of the main theorems (e.g., the isomorphism type of Aut(A_{D_5}) and Aut(A_{D_5}/Z)) with precise notation matching the body of the paper.
  2. Ensure that all citations to the prior D_n results (n≠5) are listed in the bibliography and referenced at the points where the extension is invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their positive assessment. We are pleased that the referee recognizes that the work resolves the remaining case n=5 and completes the classification of automorphisms for spherical type D_n Artin groups.

Circularity Check

0 steps flagged

No circularity; D5 case handled by direct extension of prior independent results

full rationale

The paper claims to determine Aut(A(D5)) and Aut(A(D5)/Z) by extending techniques already established for spherical Dn Artin groups when n≠5. The abstract explicitly positions the work as settling the sole remaining case in an existing classification. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse the central claim appear in the provided text. The derivation for the n=5 case is presented as a new, case-specific computation rather than a renaming or tautological restatement of prior inputs. Prior results on other Dn are treated as external context, not as an unverified self-citation chain that forces the D5 outcome by construction. This is the standard non-circular pattern for completing a classification in group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a classification result in combinatorial group theory. No numerical parameters are fitted, no new entities are postulated, and the work rests on the standard axiomatic definition of Artin groups of spherical type together with previously established results for other n.

axioms (1)
  • standard math Standard axioms and relations defining Artin groups of spherical type Dn from their Coxeter diagrams
    The paper invokes the usual presentation of the D5 Artin group and its center.

pith-pipeline@v0.9.1-grok · 5557 in / 1166 out tokens · 26832 ms · 2026-06-26T05:37:45.825554+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Korean Math

    B\,H An, Y Cho, The automorphism groups of Artin groups of edge-separated CLTTF graphs, J. Korean Math. Soc. 60 (6) (2023), 1171--1213

    2023

  2. [2]

    M\,A Blufstein, A Martin, N Vaskou, Homomorphisms between XL-type Artin groups, preprint, 2025, arXiv:2410.19091v2

    work page arXiv 2025

  3. [3]

    E Brieskorn, K Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245--271

    1972

  4. [4]

    F Castel, Repr\'esentations g\'eom\'etriques des groupes de tresses, Ph.\,D.\ Thesis, Universit\'e de Bourgogne, 2009

    2009

  5. [5]

    378 (2016)

    F Castel, Geometric representations of the braid groups, Astérisque No. 378 (2016)

    2016

  6. [6]

    F Castel, L Paris, Endomorphisms of Artin groups of type D , Algebr. Geom. Topol. 25 (2025), no 7, 3975--4008

    2025

  7. [7]

    R Charney, J Crisp, Automorphism groups of some affine and finite type Artin groups, Math. Res. Lett. 12 (2005), no. 2--3, 321--333

    2005

  8. [8]

    L Chen, K Kordek, D Margalit, Homomorphisms between braid groups, preprint, 2023, arXiv:1910.00712v2

    work page arXiv 2023

  9. [9]

    A\,M Cohen, D\,B Wales, Linearity of Artin groups of finite type, Israel J. Math. 131 (2002), 101--123

    2002

  10. [10]

    J Crisp, Automorphisms and abstract commensurators of 2-dimensional Artin groups, Geom. Topol. 9 (2005), 1381--1441

    2005

  11. [11]

    Artin groups of type B and D

    J Crisp, L Paris, Artin groups of type B and D, preprint, 2002. arXiv:math/0210438

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  12. [12]

    J Crisp, L Paris, Artin groups of type B and D , Adv. Geom. 5 (2005), no. 4, 607--636

    2005

  13. [13]

    M B Day, Peak reduction and finite presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009), 817--855

    2009

  14. [14]

    Algebra 268 (2003), no.\ 1, 39--57

    F Digne, On the linearity of Artin braid groups, J. Algebra 268 (2003), no.\ 1, 39--57

    2003

  15. [15]

    J\,L Dyer, E\,K Grossman, The automorphism groups of the braid groups, Amer. J. Math. 103 (1981), no. 6, 1151--1169

    1981

  16. [16]

    Group Theory 3 (2000), no

    N\,D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000), no. 2, 213--223

    2000

  17. [17]

    arXiv:2407.19940

    J Huang, D Osajda, N Vaskou, Rigidity and classification results for large-type Artin groups, preprint, 2024. arXiv:2407.19940

    work page arXiv 2024

  18. [18]

    Algebra 701 (2026), 436--476

    O Jones, G Mangioni, G Sartori, A combination theorem for the twist conjecture for Artin groups, J. Algebra 701 (2026), 436--476

    2026

  19. [19]

    London Math

    M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. (2) 52 (1995), no.\ 2, 318--334

    1995

  20. [20]

    S\,Y Orevkov, Homomorphisms of commutator subgroups of braid groups with small number of strings, Ann. Fac. Sci. Toulouse Math. (6) 33 (2024), no. 1, 105--121

    2024

  21. [21]

    Algebra 678 (2025), 809--830

    L Paris, I Soroko, Endomorphisms of Artin groups of type A_ n , J. Algebra 678 (2025), 809--830

    2025

  22. [22]

    Algebra 678 (2025), 831--861

    L Paris, I Soroko, Endomorphisms of Artin groups of type B_ n , J. Algebra 678 (2025), 831--861

    2025

  23. [23]

    Algebra 126 (1989), no.\ 1, 34--60

    H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), no.\ 1, 34--60

    1989

  24. [24]

    300 (2021), Paper No.\ 107770, 15 pp

    I Soroko, Artin groups of types F_4 and H_4 are not commensurable with that of type D_4 , Topology Appl. 300 (2021), Paper No.\ 107770, 15 pp

    2021

  25. [25]

    Dedicata 219 (2025), Paper No.\ 16, 20 pp

    N Vaskou, Automorphisms of large-type free-of-infinity Artin groups, Geom. Dedicata 219 (2025), Paper No.\ 16, 20 pp

    2025