On the Weak Right Order of a Right-Angled Coxeter System
Pith reviewed 2026-06-26 13:04 UTC · model grok-4.3
The pith
The ancestor property holds for all non-identity elements of right-angled Coxeter systems.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In any Coxeter system, every non-identity fully commutative element has the ancestor property. Since every element of a right-angled Coxeter system is fully commutative, every non-identity element of such a system has the ancestor property. Right-angled Coxeter systems are precisely the reflection systems equipped with a reflection cocycle that obeys the meet intersection condition.
What carries the argument
The ancestor property of an element w, which asserts a unique non-trivial involution of maximal length inside the weak right interval [1,w].
If this is right
- The ancestor property holds for every non-identity element in any right-angled Coxeter system.
- The ancestor property holds for every non-identity fully commutative element in an arbitrary Coxeter system.
- Right-angled Coxeter systems are axiomatized as reflection systems whose reflection cocycles satisfy the meet intersection condition.
Where Pith is reading between the lines
- The axiomatization may be used to verify the ancestor property by direct computation in concrete right-angled examples.
- The result indicates that the ancestor property should be checked in other families of Coxeter systems containing large sets of fully commutative elements.
- The meet intersection condition on cocycles may admit independent study as a structural feature of reflection systems.
Load-bearing premise
Every element of a right-angled Coxeter system is fully commutative.
What would settle it
A right-angled Coxeter system containing a non-identity element w whose weak right interval contains two or more distinct non-trivial involutions of the same maximal length.
read the original abstract
Let $ (W,S)$ be a Coxeter system, and let $ w\in W$. Let $ [1,w] := \{ x\in W \mid x \leq_{R} w \} $ where $ \leq_{R}$ denotes the weak right order of $ (W,S)$. The element $ w$ is said to have the \emph{ancestor property} if there is a unique non-trivial involution of maximal length in the set $ [1,w]$. The ancestor property was first defined by Hart and Rowley in \cite{hart2025noteinvolutionprefixescoxeter} where they conjectured that all non-identity elements in a finite Coxeter system have the ancestor property. In an arbitrary Coxeter system $(W,S)$, we show that the ancestor property holds for any non-identity fully commutative element (see \cite{stembridge1996fully} for the definition of a fully commutative element). In particular, since any element of a right-angled Coxeter system is fully commutative, we show that the ancestor property holds for all non-identity elements of a right-angled Coxeter system. Lastly, we also provide an axiomatization of right-angled Coxeter systems as reflection systems with a reflection cocycle that obeys a certain property called the \emph{meet intersection condition}.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that the ancestor property holds for every non-identity fully commutative element w in an arbitrary Coxeter system (W,S) with respect to the weak right order. Because every element of a right-angled Coxeter system is fully commutative, the result specializes to show that all non-identity elements of a right-angled Coxeter system have the ancestor property. The manuscript also supplies an axiomatization of right-angled Coxeter systems as reflection systems whose reflection cocycle satisfies the meet intersection condition.
Significance. If the central argument is correct, the work settles the Hart–Rowley conjecture for the entire class of right-angled Coxeter systems and supplies a new reflection-theoretic characterization of that class. The reduction to the fully commutative case is the load-bearing step; the right-angled specialization follows immediately from the standard fact (Stembridge 1996) that right-angled systems have only commutation relations.
minor comments (2)
- The statement that every element of a right-angled Coxeter system is fully commutative is invoked without an explicit reference or short proof; adding a one-sentence reminder citing Stembridge (1996) would improve readability.
- The axiomatization in the final section is presented after the main theorem; a brief discussion of whether the meet-intersection condition is used in the proof of the ancestor property, or is independent, would clarify the logical dependence.
Simulated Author's Rebuttal
We thank the referee for their positive and supportive report, which recommends minor revision and notes that the central argument settles the Hart–Rowley conjecture for right-angled Coxeter systems if correct. No major comments were listed in the report, so we provide no point-by-point responses below.
Circularity Check
No significant circularity; derivation relies on external definitions and prior results
full rationale
The paper proves the ancestor property for non-identity fully commutative elements in arbitrary Coxeter systems, then specializes to right-angled systems via the known fact (cited to Stembridge 1996) that all elements there are fully commutative by definition of the braid relations. This specialization is immediate from the definition and introduces no self-referential reduction or fitted parameter. The cited conjecture is from unrelated authors (Hart-Rowley). The final axiomatization of right-angled systems is presented separately and does not collapse the main claim. No equations, self-citations, or ansatzes in the provided abstract or described claims reduce the result to its inputs by construction. The derivation is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard axioms and exchange/deletion properties of Coxeter systems and their weak orders
- domain assumption Every element of a right-angled Coxeter system is fully commutative
Reference graph
Works this paper leans on
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[1]
Bj ¨orner and F
A. Bj ¨orner and F. Brenti.Combinatorics of Coxeter groups, volume 231. Springer, 2005
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[2]
M. Dyer. Reflection subgroups of coxeter systems.Journal of Algebra, 135(1):57–73, 1990
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[3]
M. J. Dyer.Hecke Algebras and Reflections in Coxeter Groups. PhD thesis, University of Sydney, 1987
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[4]
S. B. Hart and P. J. Rowley. A note on involution prefixes in coxeter groups, 2025
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[5]
J. E. Humphreys.Reflection groups and Coxeter groups. Number 29. Cambridge university press, 1992
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[6]
J. R. Stembridge. On the fully commutative elements of coxeter groups.Journal of Algebraic Combinatorics, 5(4):353–385, 1996. (H. Gimenez)Department of Mathematics, B26 Hayes-Healy Building, University of Notre Dame, Indiana 46556, U.S.A. Email address:hgimenez@nd.edu
1996
discussion (0)
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