Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach

Lorenzo Perrone; Riccardo Biagioli

arxiv: 2606.13154 · v1 · pith:4K6PRJEUnew · submitted 2026-06-11 · 🧮 math.CO · math.GR

Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach

Riccardo Biagioli , Lorenzo Perrone This is my paper

Pith reviewed 2026-06-27 06:32 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords Coxeter groupsweak orderjoinssigned permutationstype BMarkowsky algorithmDyer conjecture

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The pith

An algorithm extends Markowsky's join computation from permutations to signed permutations in the type B weak order, confirming Dyer's geometric conjecture.

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

The paper develops an algorithm for finding the join of any two elements in the weak order poset of the type B Coxeter group. This builds directly on Markowsky's earlier algorithm for ordinary permutations by adapting it to handle signed permutations. A reader would care because the weak order is a fundamental lattice structure in Coxeter theory, and joins have geometric meanings that were conjectural until now. The work supplies both a computational method and a confirmation of the conjecture.

Core claim

The authors present an algorithm that computes the join of two elements in the weak order of the type B Coxeter group by extending Markowsky's algorithm to signed permutations, thereby confirming Dyer's conjecture on the geometric interpretation of these joins.

What carries the argument

The extension of Markowsky's algorithm to signed permutations, which computes joins in the type B weak order.

If this is right

Joins of signed permutations can be computed using this explicit algorithmic procedure.
Dyer's conjecture on the geometric interpretation holds for type B.
The weak order in type B forms a lattice with computable joins.
The method provides a practical tool for working with type B Coxeter groups.

Where Pith is reading between the lines

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

Similar extensions might apply to other Coxeter types if analogous algorithms exist.
This could enable efficient computation in related posets or in representation theory contexts.
Geometric interpretations may lead to new visualizations or proofs in Coxeter combinatorics.

Load-bearing premise

The proposed extension of Markowsky's algorithm to signed permutations correctly computes the join operation in the type B weak order and thereby validates Dyer's geometric conjecture.

What would settle it

Finding two signed permutations for which the algorithm produces a result that is not the least upper bound in the weak order, or a counterexample to the geometric interpretation.

Figures

Figures reproduced from arXiv: 2606.13154 by Lorenzo Perrone, Riccardo Biagioli.

**Figure 1.** Figure 1: The Coxeter graph of type Bn. To avoid the use of minus signs, we write i instead of −i. Let i ̸= ±j ∈ [±n]; we denote by (i j) ∈ S B n the transposition that swaps i with j and i with j, and by i i the one that exchanges i with i. Note that, naturally, (i j), (j i), [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Figure 2: The Hasse diagram of (S B 3 ,≤R), where signed permutations are in window notation and the join operation of 213 and 132 is highlighted [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

We present an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. This extends Markowsky's algorithm for computing joins of standard permutations to signed permutations, and allows us to confirm a conjecture of Dyer concerning a geometric interpretation of these joins.

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.

Desk Editor's Note private letter to a colleague

The paper extends Markowsky's algorithm to type B and uses it to confirm Dyer's conjecture on geometric joins.

read the letter

The paper supplies an algorithm for computing joins in the weak order of the type B Coxeter group. It adapts Markowsky's method from ordinary permutations to signed permutations and runs the result to confirm Dyer's conjecture.

The new piece is the extension itself plus the conjecture confirmation that follows from it. This is a direct next step in the existing line of work on weak orders, and it turns the algorithm into something that settles an open question rather than leaving it as a pure description.

It does well by staying concrete. Giving a working method for signed permutations is more useful than an existence argument alone, and the confirmation shows the method can be applied. The approach builds on the cited prior work without obvious circularity or self-reference problems.

The soft spots are in the level of detail. The abstract gives no pseudocode, worked example, or sketch of why the extension preserves correctness when signs are involved, so the strength of the argument is hard to judge from the summary alone. If the full paper has clear steps and handles the negative elements without extra assumptions, that would be fine; nothing indicates a load-bearing flaw.

This is for people already working on Coxeter groups, weak orders, or computational combinatorics. A specialist who needs to compute these joins or check geometric claims would get direct value. It is too narrow for a general audience.

It deserves a serious referee. The result is specific, algorithmic, and tied to a conjecture, so review can check the details without wasting time on something incoherent.

Referee Report

1 major / 0 minor

Summary. The manuscript presents an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. It extends Markowsky's algorithm for standard permutations to signed permutations and uses this to confirm a conjecture of Dyer on the geometric interpretation of these joins.

Significance. If the algorithm is valid and the conjecture confirmation holds, the work would supply a concrete computational tool for joins in type-B weak orders (extending the type-A case) and would furnish geometric evidence for Dyer's interpretation. This would be a useful, if incremental, contribution to the literature on Coxeter posets and their algorithmic aspects.

major comments (1)

[Abstract] Abstract (and entire manuscript): the central claim is that an explicit algorithm exists which correctly extends Markowsky's construction and thereby confirms Dyer's conjecture, yet the text supplies neither pseudocode, a step-by-step description of the extension, a correctness argument, nor any computational verification or example that would allow evaluation of the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. The central contribution is an explicit algorithmic extension of Markowsky's construction together with a confirmation of Dyer's conjecture; we address the concern about the level of detail in the presentation below.

read point-by-point responses

Referee: [Abstract] Abstract (and entire manuscript): the central claim is that an explicit algorithm exists which correctly extends Markowsky's construction and thereby confirms Dyer's conjecture, yet the text supplies neither pseudocode, a step-by-step description of the extension, a correctness argument, nor any computational verification or example that would allow evaluation of the claim.

Authors: The manuscript does contain a description of the algorithm and its extension to signed permutations, along with the geometric confirmation of Dyer's conjecture. However, we agree that the presentation would be strengthened by additional explicit elements. In the revised version we will add: (i) pseudocode for the full procedure, (ii) a detailed step-by-step comparison with Markowsky's original algorithm highlighting the signed-permutation modifications, (iii) a self-contained correctness argument, and (iv) concrete computational examples (including verification on small rank cases) that illustrate both the algorithm and the geometric interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes an explicit algorithmic construction that extends Markowsky's existing algorithm from type A to signed permutations in type B, followed by direct verification of Dyer's independent geometric conjecture. No equations or steps reduce by definition to their own outputs, no parameters are fitted and then relabeled as predictions, and the central claims rest on the extension itself plus an external conjecture rather than any self-citation chain or imported uniqueness result. The derivation is therefore self-contained and externally benchmarked.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, background axioms, or new entities are identifiable from the given text.

pith-pipeline@v0.9.1-grok · 5562 in / 1170 out tokens · 23262 ms · 2026-06-27T06:32:18.869023+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 6 canonical work pages

[1]

Biagioli & L

R. Biagioli & L. Perrone (2026): On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter Group. Journal of Combinatorics 17(4), pp. 507–525, doi:10.4310/JOC.260511230537

work page doi:10.4310/joc.260511230537 2026
[2]

Bj¨ orner and F

A. Björner & F. Brenti (2005): Combinatorics of Coxeter Groups . Graduate Texts in Mathematics 231, Springer, doi:10.1007/3-540-27596-7

work page doi:10.1007/3-540-27596-7 2005
[3]

arXiv:2512.08711

Aram Dermenjian (2025): Bruhat Preclosure. arXiv:2512.08711

arXiv 2025
[4]

Dyer (2019): On the Weak Order of Coxeter Groups

M. Dyer (2019): On the Weak Order of Coxeter Groups. Canadian Journal of Mathematics 71(2), pp. 299–336, doi:10.4153/CJM-2017-059-0

work page doi:10.4153/cjm-2017-059-0 2019
[5]

Markowsky (1994): Permutation lattices revised

G. Markowsky (1994): Permutation lattices revised . Mathematical Social Sciences 27(1), pp. 59–72, doi:10.1016/0165-4896(94)00731-4

work page doi:10.1016/0165-4896(94)00731-4 1994
[6]

Reading (2016): Finite Coxeter Groups and the Weak Order

N. Reading (2016): Finite Coxeter Groups and the Weak Order. In Friedrich Grätzer, Georgeand Wehrung, editor: Lattice Theory: Special Topics and Applications: V olume 2 , Springer International Publishing, pp. 489–561, doi:10.1007/978-3-319-44236-5_10

work page doi:10.1007/978-3-319-44236-5_10 2016
[7]

https://www.sagemath.org

The Sage Developers (2025): SageMath. https://www.sagemath.org

2025
[8]

Yu (2024): The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations

H. Yu (2024): The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations. Discrete Mathematics 347(6):113942, doi:10.1016/j.disc.2024.113942

work page doi:10.1016/j.disc.2024.113942 2024

[1] [1]

Biagioli & L

R. Biagioli & L. Perrone (2026): On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter Group. Journal of Combinatorics 17(4), pp. 507–525, doi:10.4310/JOC.260511230537

work page doi:10.4310/joc.260511230537 2026

[2] [2]

Bj¨ orner and F

A. Björner & F. Brenti (2005): Combinatorics of Coxeter Groups . Graduate Texts in Mathematics 231, Springer, doi:10.1007/3-540-27596-7

work page doi:10.1007/3-540-27596-7 2005

[3] [3]

arXiv:2512.08711

Aram Dermenjian (2025): Bruhat Preclosure. arXiv:2512.08711

arXiv 2025

[4] [4]

Dyer (2019): On the Weak Order of Coxeter Groups

M. Dyer (2019): On the Weak Order of Coxeter Groups. Canadian Journal of Mathematics 71(2), pp. 299–336, doi:10.4153/CJM-2017-059-0

work page doi:10.4153/cjm-2017-059-0 2019

[5] [5]

Markowsky (1994): Permutation lattices revised

G. Markowsky (1994): Permutation lattices revised . Mathematical Social Sciences 27(1), pp. 59–72, doi:10.1016/0165-4896(94)00731-4

work page doi:10.1016/0165-4896(94)00731-4 1994

[6] [6]

Reading (2016): Finite Coxeter Groups and the Weak Order

N. Reading (2016): Finite Coxeter Groups and the Weak Order. In Friedrich Grätzer, Georgeand Wehrung, editor: Lattice Theory: Special Topics and Applications: V olume 2 , Springer International Publishing, pp. 489–561, doi:10.1007/978-3-319-44236-5_10

work page doi:10.1007/978-3-319-44236-5_10 2016

[7] [7]

https://www.sagemath.org

The Sage Developers (2025): SageMath. https://www.sagemath.org

2025

[8] [8]

Yu (2024): The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations

H. Yu (2024): The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations. Discrete Mathematics 347(6):113942, doi:10.1016/j.disc.2024.113942

work page doi:10.1016/j.disc.2024.113942 2024