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arxiv: 2605.22065 · v1 · pith:7ZISS3T3new · submitted 2026-05-21 · 🧮 math.GR

Symmetry and Rigidity of Star-Shaped Coxeter Systems

Arijit Mahato , Tushar Kanta Naik , A Rameswar Patro This is my paper

Pith reviewed 2026-05-22 02:55 UTC · model grok-4.3

classification 🧮 math.GR MSC 20F55
keywords Coxeter groupsautomorphism groupsstar-shaped diagramsR-infinity propertypartial conjugationstransvectionsrigidityisomorphism problem
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0 comments X

The pith

For Coxeter groups with star-shaped finite diagrams, every automorphism decomposes as a product of inner automorphisms, diagram automorphisms, transvections, and two families of partial conjugations.

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

The paper examines Coxeter groups whose defining diagrams are finite and star-shaped. It supplies an explicit breakdown of every automorphism into a combination of four basic kinds of maps. With this breakdown in hand the authors establish that the groups satisfy the R-infinity property and that rigidity results follow, which in turn solve the isomorphism problem inside the class. A reader would care because the geometric restriction on the diagram turns the usually opaque automorphism group into an object whose elements and consequences can be listed and used directly.

Core claim

The central claim is that if W is a Coxeter group whose Coxeter diagram is finite and star-shaped, then every automorphism of W can be written uniquely as a product of an inner automorphism, a diagram automorphism, a transvection, and a partial conjugation from one of two families. The same decomposition is applied to the short exact sequence relating Inn(W) to Aut(W) and Out(W). Moussong's criteria then show that W is hyperbolic and therefore possesses the R-infinity property. The resulting rigidity statements suffice to decide when two star-shaped Coxeter systems are isomorphic.

What carries the argument

The star-shaped finite Coxeter diagram, whose central vertex and attached branches limit the possible actions on the generating set and thereby force every automorphism to factor through the four listed types.

If this is right

  • The outer automorphism group Out(W) is generated by the images of diagram automorphisms together with the transvections and the two families of partial conjugations.
  • Every such W satisfies the R-infinity property, so that the number of twisted conjugacy classes is infinite for every automorphism.
  • Rigidity properties derived from the decomposition allow one to solve the isomorphism problem by comparing diagrams up to the permitted symmetries.
  • Moussong's hyperbolicity criteria combine with the star-shaped condition to guarantee the structural results without additional assumptions.

Where Pith is reading between the lines

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

  • The same factorization technique might apply to other diagrams that are trees or have bounded branching, enlarging the class of Coxeter groups whose automorphisms are fully understood.
  • The explicit generators for Aut(W) could be used to compute the action on the associated hyperbolic space and to study orbit growth or fixed-point properties.
  • Rigidity results obtained here suggest that similar diagram restrictions could yield computable invariants for related classes such as certain Artin groups or reflection groups.

Load-bearing premise

The diagram must be both finite and star-shaped; the proofs of the automorphism decomposition and the R-infinity property rest on this geometric restriction.

What would settle it

An explicit listing of all automorphisms of the Coxeter group whose diagram consists of a single central vertex joined to three length-one branches, showing even one automorphism that lies outside the four described types, would disprove the claimed decomposition.

read the original abstract

We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the natural short exact sequence $1 \to \Inn (W) \to \Aut (W) \to \Out (W) \to 1$. Using Moussong's criteria for hyperbolicity, we show that these groups possess the $R_\infty$-property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems.

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 / 3 minor

Summary. The paper claims a complete description of Aut(W) for Coxeter groups W with finite star-shaped Coxeter diagrams. It asserts that every automorphism factors as a product of inner automorphisms, diagram automorphisms, transvections, and two families of partial conjugations. The work further analyzes the short exact sequence 1 → Inn(W) → Aut(W) → Out(W) → 1, proves the R_∞ property via Moussong hyperbolicity, establishes rigidity results, and solves the isomorphism problem within the class of star-shaped Coxeter systems through case analysis on the central vertex and arms of the diagram.

Significance. If the classification and decomposition hold, the results extend known structural theorems on automorphism groups of Coxeter groups to a geometrically restricted but infinite family of diagrams. The explicit generators for Aut(W), the verification of the R_∞ property, and the resolution of the isomorphism problem constitute concrete advances that could serve as a model for similar analyses in other diagram classes. The use of direct preservation checks on Coxeter relations for the proposed maps is a strength when the case analysis is exhaustive.

minor comments (3)
  1. [Abstract and §1] The abstract states that automorphisms 'decompose as a product' of the listed types, but the introduction or §2 should explicitly state whether the decomposition is unique or canonical and how the factors commute or interact.
  2. [§3] Notation for the two families of partial conjugations is introduced without a preliminary table or diagram illustrating the action on the standard generators; adding such a figure would improve readability of the case analysis.
  3. [§4] The proof that the diagram is star-shaped implies no other automorphism types arise relies on controlling conjugacy classes; a short lemma summarizing the possible images of the central generator under any automorphism would make this step easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results on the automorphism group of star-shaped Coxeter systems, the positive assessment of their significance, and the recommendation for minor revision. We will incorporate any editorial improvements to enhance clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is a classification of Aut(W) obtained by direct case analysis on the central vertex and arms of a finite star-shaped Coxeter diagram, verifying that proposed maps preserve the Coxeter relations. The R_∞ property follows from Moussong's external hyperbolicity criteria, and rigidity/isomorphism results invoke known techniques outside the present derivation. No step reduces by definition, by fitting a parameter then relabeling it a prediction, or by a load-bearing self-citation chain whose content is unverified. The geometric restriction is an explicit hypothesis used to bound the enumeration, not smuggled in via prior work. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure proof paper in group theory. It invokes standard facts about Coxeter systems and Moussong's hyperbolicity criterion but introduces no fitted numerical parameters and no new postulated entities.

axioms (1)
  • standard math Standard axioms and relations of Coxeter systems (finite diagram, star-shaped geometry)
    Invoked throughout the description of Aut(W) and the application of Moussong's criteria.

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Reference graph

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