Computing finite Weyl groupoids

Iv\'an Angiono; Leandro Vendramin

REVIEW 3 major objections 3 minor 30 references

Computing finite Weyl groupoids

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper gives step-by-step algorithms that compute the finite generalized root system of a diagonal-type Nichols algebra or a contragredient Lie superalgebra, then derive the Lyndon words and hyperwords of the associated PBW basis.

desk verdict Printed Algorithm 3.1 has a left/right reflection-order bug that breaks on A2, but the GAP package and the underlying recipe are likely salvageable and worth engaging. read the letter →

arxiv 2505.24555 v1 pith:44EMLUNF submitted 2025-05-30 math.RT math.QAmath.RA

Iván Angiono , Leandro Vendramin This is my paper

classification math.RTmath.QAmath.RA MSC 17B3717B2220F5516T05

keywords WeylgroupoidsgeneralizedrootsystemsNicholsalgebrasofdiagonaltypecontragredientLiesuperalgebrasLyndonwordsPBWbaseshyperwordsfinite

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to make finite generalized root systems computable rather than classification-lookup. For a diagonal-type braiding matrix, or for a matrix with parity vector defining a contragredient Lie superalgebra, it gives iterative algorithms that return all positive roots, the Cartan roots, the root heights, the longest element of the Weyl groupoid, and the dimension, whenever the root system is finite. From those roots, ordered convexly, a second round of algorithms derives the unique Lyndon word for each positive root and the corresponding hyperword in the PBW basis. A sympathetic reader would care because these data are precisely the input needed to write a minimal presentation of the Nichols algebra and to read off the superdimension of the Lie superalgebra without reproducing a case-by-case classification.

What carries the argument

The machinery is the Weyl groupoid of the generalized root system together with its longest element. The load-bearing identity is (1.10): if $\omega_0^x = \sigma_{i_1}^x \cdots \sigma_{i_\ell}$ is a reduced expression, then the positive roots at $x$ are exactly $\beta_j = s_{i_1}^x \cdots s_{i_{j-1}}(\alpha_{i_j})$, so computing a reduced expression of the longest element is equivalent to listing all roots. The algorithms construct that expression by alternating reflection of the coordinate matrix with reflection of the defining data: for Nichols algebras of diagonal type the update is the reflection of the braiding matrix justified by reflection isomorphisms of Drinfeld doubles; for contragredient Lie superalgebras the update uses explicit reflection formulas for matrix and parity vector. Lyndon words, words smaller than all their proper endings, are then recovered from the convex root order by decomposing each root into two smaller positive roots.

What would settle it

Take a finite example such as the rank-three braiding matrix in Example 4.1, independently compute the length of the longest element of its Weyl groupoid from the Coxeter datum, and compare it with the length of the word Algorithm 3.1 outputs; a longer word, or a missing positive root when the roots are checked against the classification, would falsify the greedy reduced-expression claim.

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Extended reading notes

Core claim

The central claim is that the whole finite root system of either family is determined by iterated simple reflections, and that a simple greedy loop computes it. Starting from the simple roots, the algorithm maintains a matrix $M$ whose columns record root coordinates; at each step it computes the generalized Cartan matrix of the current braiding or current pair, applies the corresponding reflection to $M$, updates the braiding matrix $q$ via (2.2) or the pair $(A,p)$ via (2.4)--(2.6), and chooses as the next reflection the smallest index whose column still lies in $\mathbb{N}^\theta_0$. When the system is finite the accumulated word is a reduced expression of the unique longest element of the Weyl groupoid, and equation (1.10) then yields every positive root as a successive image of a simple root. The paper claims that the resulting root order is the convex order needed by the second stage, in which the unique Lyndon word for each root is built by taking the maximum concatenation of Lyndon words of smaller-root decompositions, and the hyperwords are obtained by braided brackets of the two parts of a minimal-length decomposition.

Load-bearing premise

The load-bearing premise is that the greedy rule, reflect at the smallest index whose column still has nonnegative entries, always produces a reduced expression of the longest element of the Weyl groupoid, so that formula (1.10) really lists every positive root, and the Lie superalgebra reflection formulas currently rest on an in-preparation manuscript.

Editorial extensions

If this is right

For any diagonal-type Nichols algebra with finite root system, the algorithms output the positive roots, their heights, a reduced expression of the longest element, and the dimension as the product of the heights.
The Lyndon words and hyperwords produced by Algorithms 3.2 and 3.3 form the PBW generators, so the output is exactly the data needed for a minimal presentation of the Nichols algebra.
For a contragredient Lie superalgebra, the restricted positive roots plus the odd non-degenerate roots determine the full positive root set through (2.7), and the count gives both components of the superdimension.
The same loop doubles as a finiteness certificate: if the accumulated number of roots exceeds the explicit bound $B_\theta$ from the classification of finite generalized root systems, the root set is infinite, so the algorithm answers finiteness without looking up the classification list.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

(Editorial inference) The greedy choice of the smallest eligible index is the one step whose correctness the paper delegates to cited results; one could test it independently by comparing the output word length with a brute-force reduced expression of the longest element on all finite rank-2 and rank-3 examples.
(Editorial inference) The Lyndon-word stage only uses a finite root system with a convex order, so it could be reused for any other family of finite root systems for which such an order can be produced.
(Editorial inference) Running the whole pipeline on all classified finite examples and matching dimensions and PBW generator degrees would provide an end-to-end consistency check of the reflection formulas and the greedy rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Plainly: the paper's central algorithm, as printed, does not work. In Algorithm 3.1 step (e), M ← (s_i^q M_1 | ... | s_i^q M_θ) is left multiplication by the reflection, which corresponds to prepending σ_i to the word, not appending it. The root recursion (1.10) requires the prefix to act on the left as s_{i_1}...s_{i_{j-1}}, and appending σ_i should update M by right multiplication, M ← M s_i^q. As printed, for Cartan type A2 the loop stalls after two steps and α_2 is never added. I checked this by hand; the stress-test note is right. The same issue appears in Algorithm 3.4 step (d).

This is a significant flaw for a paper whose contribution is explicit pseudocode. However, it looks like a presentation bug rather than a mathematical failure: the GAP package and the examples reproduce the known g(2,3) root system and heights, so the implementation probably uses the corrected order. The paper also deserves credit for making the reflection procedure explicit, for the Lyndon word recurrence based on [10], and for packaging this as a freely available, DOI-cited GAP tool. That is genuinely useful for the Nichols algebra community.

The other soft spots are less severe. The greedy minimal-index choice in step (g) is asserted without proof, though any index whose column lies in N^θ_0 should yield a reduced step by (1.9), so this is a minor gap. The termination bound via B_θ from the classification [15] is fine. More concerning is the Lie superalgebra half: Theorem 2.5 and the explicit formulas (2.4)-(2.6) are cited to the authors' own in-preparation manuscript [3]. There is no independent verification here, which is fine for a research announcement but weak for a paper whose algorithms depend on those formulas. That dependency should be stated upfront, ideally with the formulas taken from [3] clearly labeled as such.

Bottom line: this is a useful computational note for specialists in Nichols algebras and root systems, but the printed algorithms need a typo fix and the dependence on [3] needs to be explicit. I would send it to referees, but I'd tell the authors that the pseudocode must be corrected before final acceptance. If you use it, test the A2 case first. For my own work, I wouldn't cite it until the pseudocode is fixed.

Referee Report

3 major / 3 minor

Summary. The paper presents algorithms for computing the finite generalized root system of a Nichols algebra of diagonal type and of a contragredient Lie superalgebra. The algorithms are designed to return the positive roots, a reduced expression of the longest element of the Weyl groupoid, the PBW data (including Lyndon words and hyperwords) for the Nichols algebra case, and the odd non-degenerate roots for the Lie superalgebra case. The authors state that the algorithms are implemented in GAP and illustrate them on the rank-three example g(2,3), where the output matches known data. The theoretical part reviews Weyl groupoids, generalized root systems, and the standard results of Heckenberger, and then introduces the Lie superalgebra construction based on results from an in-preparation manuscript [3].

Significance. If the algorithms are correct, they provide useful computational tools that avoid consulting the full classifications [15,17] and that output additional structural data such as Lyndon words, PBW generators, dimensions, and the longest element. The Nichols algebra part is grounded in independent theorems of Heckenberger [16,18], and the authors make the code freely available with a Zenodo DOI, which should facilitate independent checking. The Lie superalgebra part, however, depends on formulas and a theorem from the unpublished source [3], and the main algorithmic loop in the printed version contains a matrix-update error. These issues affect the correctness of the central claims as written, although they appear locally fixable.

major comments (3)

[§3.1, Algorithm 3.1, step (e); §3.2, Algorithm 3.4, step (d)] The update M ← (s_i M_1 | ... | s_i M_θ) is a left multiplication by the current reflection. If, as equation (1.10) indicates, M is meant to record the cumulative reflection applied to the initial basis after the prefix w = σ_{i_1}...σ_{i_t}, then appending σ_i in step (i) changes the prefix to w σ_i and requires M ← M s_i. The printed left multiplication corresponds to prepending σ_i and therefore yields the wrong M. This is not a notational subtlety: for the Cartan type A2 braiding, after two iterations the algorithm has M = s_2 s_1 and no admissible index k ≠ i in step (g), although the loop condition holds, so the positive root α_2 is never added. The same defect breaks Algorithm 3.4. Because this is the core loop of both algorithms, the algorithms as printed do not compute the claimed root sets.
[§2.2, Theorem 2.5 and equations (2.4)-(2.6)] The Lie superalgebra construction and the claimed generalized root system are taken from [3], a manuscript listed as 'In preparation' by N. Andruskiewitsch and I. Angiono. Algorithm 3.4 and Example 4.2 depend directly on these formulas and on Theorem 2.5; no proof or independent verification is supplied here. Until [3] is available or the relevant statements are proved, the Lie superalgebra half of the paper is conditional. I request that the authors either include the proofs in this paper or explicitly state the dependence and mark the affected results as conditional.
[§3.1, Algorithm 3.1, step (g) and termination] The greedy rule i ← min{k ≠ i : M_k ∈ N^θ_0} is asserted to produce a reduced expression of the longest element, but no proof is given. Equation (1.9) only shows that any such k gives a reduced step of length t+1; one still has to prove that the process never reaches a state with length < ℓ(ω_0) and no admissible k. The termination sentence citing [15] bounds the size of finite root systems but does not justify the greedy construction. The same gap applies to Algorithm 3.4. Please provide a proof, or a clear reference where this specific greedy statement is proved.

minor comments (3)

[Example 1.1 and §2.1] There are small typographical errors: 'grupoid' in Example 1.1 and 'maximal lenght' in §2.1 should be corrected.
[Algorithm 3.2, step (3)(c)(i)] After reordering Δ_+ by height, the equality γ_j = β_k needs the correspondence between the two orderings to be stated; as written, the two indices refer to different enumerations and this is confusing.
[Section 2.1] The statement that 'the height is always finite' for finite root systems is terse; please make explicit that 'height' here means the order N_β = ord q_β, not the combinatorial height of the root.

Circularity Check

1 steps flagged · score 4.0 of 10

Nichols-algebra root computation is self-contained; the Lie-superalgebra half rests on a load-bearing self-citation to the authors' in-preparation [3].

self citation load bearing [Section 2.2, formulas (2.4)-(2.6) and Theorem 2.5; used by Algorithm 3.4 in Section 3.2]
""The notation and results in this Subsection are mainly from [3]." ... "Theorem 2.5.[3] (C (A,p),(∆ (B,q))(B,q)∈X (A,p) )is a generalized root system." Reference [3]: "N. Andruskiewitsch and I. Angiono. Generalized root systems, contragredient Lie superalgebras and Nichols algebras. In preparation, page 155 pp.""

Algorithm 3.4 is offered as computing the positive roots and longest word of a contragredient Lie superalgebra. Its update steps use c^{(A,p)}_{ij} from (2.4)-(2.5), the reflection matrices ρ_i(A,p) from (2.6), and Theorem 2.5, which asserts that (C,∆) constructed this way is a generalized root system. All of these are attributed only to [3], an in-preparation manuscript whose authors include Angiono, an author of the present paper. No proof is reproduced here, and [3] provides no machine-checked or otherwise independently verifiable justification in the text. The superalgebra half of the advertised computation therefore rests on an unverified self-citation rather than on an argument in this paper.

full rationale

For the Nichols-algebra branch, the derivation is not circular: the positive roots, reduced expressions, and Lyndon words are obtained from Heckenberger's published theorems ([16], [18]), the Heckenberger-Yamane Coxeter-groupoid results ([21]), and the Cuntz-Heckenberger classification ([15]); no parameter is fitted to the output and the computation is self-contained against external benchmarks. The only circularity-level defect is the Lie-superalgebra branch, where the root-system structure (formulas (2.4)-(2.6) and Theorem 2.5) is imported from [3], an in-preparation paper by Andruskiewitsch and Angiono, with Angiono as a coauthor of this paper. That citation is load-bearing for Algorithm 3.4 and is not independently verifiable in the present text, so the paper is only partially self-citational rather than globally circular. The apparent left-versus-right multiplication issue in Algorithm 3.1 step (e) is a correctness bug candidate, not a circular-reduction, so it does not affect this score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The algorithms rest on the established theory of Weyl groupoids, Nichols algebras, and contragredient Lie superalgebras, plus the classification of finite Weyl groupoids for termination. No free parameters are fitted. No new entities are invented. The main unverified dependency is [3], an in-preparation manuscript used for the Lie superalgebra root system theorem and reflection formulas.

assumptions (4)

domain assumption The classification of finite Weyl groupoids [15] bounds the number of positive roots by B_theta, used as the termination criterion.
Algorithms 3.1 and 3.4 stop when the root count exceeds max(250, theta^2), citing [15].
domain assumption Lusztig-type isomorphisms T_i exist for Nichols algebras of diagonal type (Theorem 2.1 from [18]).
Used to move between q and rho_i(q) in Algorithm 3.1 and to derive PBW generators.
domain assumption For contragredient Lie superalgebras, the collection (C^{(A,p)}, (Delta^{(B,q)})) is a generalized root system (Theorem 2.5 from [3]).
Justifies that the set computed by Algorithm 3.4 is the root system; [3] is an in-preparation manuscript.
standard math The PBW basis theorem of Kharchenko [25] and the uniqueness and recurrence of Lyndon words [10] hold.
Algorithms 3.2 and 3.3 compute Lyndon words and hyperwords from this theory.

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Cite this review

Pith. "Pith review of Computing finite Weyl groupoids." pith.science (2026). https://pith.science/paper/44EMLUNF

@misc{pith2026250524555,
  author       = {Pith},
  title        = {Pith review of: Computing finite Weyl groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44EMLUNF}},
  note         = {Machine review of arXiv:2505.24555}
}

read the original abstract

We present algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. As a consequence, we obtain an algorithm to compute the Lyndon words in the Kharchenko PBW basis associated to each positive root, along with their corresponding hyperwords. This data is essential for obtaining a minimal presentation of Nichols algebras of diagonal type with a finite root system.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [10]

I. E. Angiono. A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems.J. Eur. Math. Soc. (JEMS), 17(10):2643–2671, 2015

work page 2015

[2] [15]

Cuntz and I

M. Cuntz and I. Heckenberger. Finite Weyl groupoids.J. Reine Angew. Math., 702:77–108, 2015

work page 2015

[3] [3]

Andruskiewitsch and I

N. Andruskiewitsch and I. Angiono. Generalized root systems, contragredient Lie superalge- bras and Nichols algebras.In preparation, page 155 pp

work page

[4] [1]

GAP – Groups, Algorithms, and Programming, Version 4.4.12

The GAP Group, 2006. GAP – Groups, Algorithms, and Programming, Version 4.4.12. Avail- able athttp://www.gap-system.org

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Andruskiewitsch

N. Andruskiewitsch. An introduction to Nichols algebras. InQuantization, geometry and noncommutative structures in mathematics and physics. Contributions of the summer school, Villa de Leyva, Columbia, pages 135–195. Cham: Springer, 2017

work page 2017

[6] [4]

Andruskiewitsch and I

N. Andruskiewitsch and I. Angiono. On finite dimensional Nichols algebras of diagonal type. Bull. Math. Sci., 7(3):353–573, 2017

work page 2017

[7] [5]

Andruskiewitsch, I

N. Andruskiewitsch, I. Heckenberger, and H. Schneider. The Nichols algebra of a semisimple Yetter-Drinfeld module.Amer. J. Math., 132(6):1493–1547, 2010

work page 2010

[8] [6]

Andruskiewitsch and H.-J

N. Andruskiewitsch and H.-J. Schneider. Finite quantum groups and Cartan matrices.Adv. Math., 154(1):1–45, 2000

work page 2000