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Search for a basis of the Temperley-Lieb algebra, using rewriting systems

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Pith's one-line read Rewriting rules terminate at the Jones normal forms for the Temperley-Lieb algebra, and the same strategy, run in a presented category, produces bases for the oriented analogue.

desk verdict A genuinely useful rewriting-system result for TL_n(δ), but the oriented algebra basis advertised in the abstract rests on a conjecture the paper explicitly leaves open. read the letter →

arxiv 2508.19360 v1 pith:7BTTRJDH submitted 2025-08-26 math.RT

Julien Thiebaut This is my paper · ORCID
classification math.RT MSC 16S1518M0568Q42
keywords Temperley-LiebalgebrarewritingsystemJonesnormalformbasisconstructionorientedcategoryconvergentpresentedmonoidal
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's project is to find bases for Temperley-Lieb algebras algorithmically, by setting up directed replacement rules and proving they always stop at a unique irreducible word. For the classical Temperley-Lieb algebra TL_n(δ), it exhibits six families of rewrite rules, proves the system convergent, and then shows by an explicit algorithm that the unique normal forms are exactly the Jones normal forms—so the normal forms are a basis. For the oriented Temperley-Lieb algebra, it shifts the setting to a presented strict monoidal category with caps, cups, and an exchange relation; there the same rewriting strategy is again convergent, so the normal forms of morphisms form a basis of each Hom(v,w). This matters because it replaces a basis construction usually done by combinatorial enumeration with a general mechanism: once convergence is proved, unique normal forms exist automatically and a finite rule list computes them.

What carries the argument

The machinery is a rewriting system: a set of words (or morphisms) equipped with directed rules that replace one subword by another. The paper proves convergence by checking termination, via a lexicographic order on words or a decreasing count of generators, and confluence, by examining critical pairs where rules overlap; a standard lemma then upgrades local confluence plus termination to full confluence. In the unoriented case, the critical-pair analysis forces two extra rules (5) and (6) that make the system confluent, and the normal forms coincide with Jones normal forms, i.e. products of antidiagonal blocks (e_i e_{i-1} ... e_j). In the oriented-category case, the key additional datum is

What would settle it

Take n=4 or 5 and exhaustively apply the rules of Theorem 2.18 to all words up to a bounded length; if any word has two different reduction paths ending in two different irreducible words, the system is not confluent and the normal forms do not give a unique basis. For the oriented category, spelling out End(2) in TL_O(q) explicitly and checking whether it is isomorphic to the corresponding oriented algebra would test the identification cited in Proposition 4.12 and conjectured in Remark 4.14.

Watch

Extended reading notes

Core claim

The central claim is that basis questions for Temperley-Lieb algebras can be settled by proving a rewriting system convergent. In the unoriented case (Theorem 2.18), the rules move δ past e_i, send e_i^2 to δe_i, collapse e_i e_{i±1} e_i to e_i, swap distant generators, and add two longer collapsing rules needed after completion; the system terminates and is locally confluent, hence convergent. Theorem 2.19 then gives a rewriting algorithm whose output is always in Jones normal form—a product of antidiagonal words (e_i e_{i-1} ... e_j)—so the unique normal forms are precisely the Jones normal forms and therefore a basis of TL_n(δ). In the oriented case (Theorem 4.15), the author presents the

Load-bearing premise

The step that carries the whole conclusion is the claim that self-maps of n in the Temperley-Lieb category match the Temperley-Lieb algebra (cited from [Abr09] without proof here), plus the conjectural matching of the oriented category's morphism spaces with the oriented algebra; if those identifications fail, the normal forms still give bases of morphism spaces but not of the algebras.

Editorial extensions

If this is right

  • Every element of TL_n(δ) has a unique normal form, computed by the six rule families; the normal forms are exactly the Jones normal forms, so they give an explicit basis without enumerating diagrams or paths.
  • The same rewriting argument gives a constructive proof that the presented algebra and the diagrammatic algebra are isomorphic: the map e_i ↦ E_i sends a basis to a basis.
  • For the oriented Temperley-Lieb category TL_O(q), each morphism space Hom(v,w) has a finite basis of irreducible morphisms, and this basis is obtained by a terminating rule set rather than by combinatorial counting.
  • If the conjectural identification of the category's morphism spaces with the oriented Temperley-Lieb algebra is proved, the category rewriting system yields an algebra basis for TLOn,k(q) automatically.
  • The convergence proof also produces a decision procedure for equality in the algebra: two words are equal exactly when their normal forms agree.

Reading between the lines

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

  • The same 'presented category plus rewriting' template should apply to other diagrammatic algebras (braid, Hecke, BMW) whenever they admit a finite monoidal presentation; the main obstacle would be finding the right exchange relation or modulo rules.
  • A natural next experiment is to fill the gap in Remark 4.14: prove that morphisms between words with exactly k occurrences of ∨ form the oriented algebra; if true, the category basis becomes an algebra basis and the oriented analogue of Theorem 2.8 follows by the same argument.
  • The author's warning that the free presented-category construction was not fully checked suggests the most fragile point is not confluence but the completeness of the presentation; a rigorous proof of the freeness or completeness of Definition 4.7 would harden the whole approach.
  • Since the category normal forms differ from Jones normal forms under the natural embedding, the oriented case likely has its own normal-form combinatorics, which could be mined for a direct combinatorial description of the basis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper, an internship report in mathematics, pursues the Temperley-Lieb algebra and its oriented analogue from the perspective of rewriting systems. In the first part, it defines TL algebra both diagrammatically and by generators and relations, recounts the classical Jones normal form, and then proposes a convergent rewriting system for the monoid generating TL_n(δ) (Theorem 2.18), with rules (1)–(6), whose normal forms it claims coincide with Jones normal forms (Theorem 2.19, via an algorithm). This yields a basis for TL_n(δ). In the second part, the paper introduces the oriented Temperley-Lieb algebra TLO_{n,k}(q), then moves to a presented strict monoidal category, the Temperley-Lieb category TL(δ) and the oriented category TLO(q). Theorem 4.15 claims that a rewriting system on TLO(q), with rules (1-left/right), (2-left/right), (3), (4), is convergent modulo the exchange relation. The paper concludes that the normal forms of this system give a basis of each Hom-space, and, via Remark 4.14, that this gives a basis of the oriented Temperley-Lieb algebra.

Significance. The non-oriented part demonstrates an original algorithmic and rewriting-based route to a known basis (Jones normal form) of the Temperley-Lieb algebra. The oriented part aims to do the same for the oriented Temperley-Lieb algebra, a subject with recent activity (e.g., Bowman et al.). The explicit convergence statement for the oriented category rewriting system (Theorem 4.15) is a potentially useful and clean result, and the visual rewriting rules in the category setting are promising. However, the paper's strongest advertised claim—that this yields a basis of the oriented Temperley-Lieb algebra TLO_{n,k}(q)—is not established: it rests on an identification (Remark 4.14) that the author explicitly leaves as a conjecture, and on a category construction whose completeness is acknowledged as unproved (§4.2, Remark 4.8). The manuscript is also candid about these gaps, which is a virtue, but the central theorem as stated in the abstract is therefore stronger than what is proved. If the conjectural identifications are supplied, the approach would provide a genuinely useful basis theorem for TLO_{n,k}(q).

major comments (4)
  1. [§4.4 and Remark 4.14] The conclusion that normal forms of the convergent rewriting system of Theorem 4.15 form a basis of the oriented Temperley-Lieb algebra TLO_{n,k}(q) requires the identification Hom_{TLO(q)}(v,w) ≅ TLO_{n,k}(q) for v,w ∈ L_k. This identification is stated in Remark 4.14 only as 'everything suggests', with no proof. Without it, Theorem 4.15 yields bases of Hom-spaces in the presented category, not a basis of the algebra. This is a load-bearing missing link for the paper's central claim.
  2. [§4.2, Definition 4.7 and Remark 4.8] The presented strict monoidal category construction underlying all category-theoretic conclusions is admittedly incomplete. The author states in §4.2 that the proofs 'haven't actually' been done, that the construction 'may lack completeness', and Remark 4.8 says the bifunctor property 'should be shown'. Since the confluence proof of Theorem 4.15 is formulated inside this construction, a rigorous proof that the construction is indeed a well-defined strict monoidal linear category is needed before the Hom-space basis claim is fully justified.
  3. [§4.3, Proposition 4.12] The non-oriented identification End_{TL(δ)}(n) ≅ TL_n(δ) is discharged entirely by a citation to [Abr09] with no detailed argument. This is acceptable as background if the cited reference is standard, but the paper then transfers this identification to the oriented setting (Remark 4.14) without an analogous cited proof. At minimum, the author should state precisely which theorem in [Abr09] gives the endomorphism-algebra isomorphism, and explain how the presentation used here (with exchange relation) matches the standard Temperley-Lieb category.
  4. [§3.2 and §4.4] The paper motivates the categorical approach by abandoning the direct word-rewriting system for TLO_{n,k}(q), but it does not clearly state the relation between the category TLO(q) and the algebra TLO_{n,k}(q). The final sentence of §4.4 says normal forms obtained in the category are not Jones normal forms for the non-oriented case, which is fine, but the reader is left without a precise functor or isomorphism linking Hom-sets of TLO(q) to the algebra. This gap is directly tied to the missing proof of Remark 4.14.
minor comments (6)
  1. [Abstract] The abstract claims rewriting 'easily obtain[s] a basis' for the oriented algebra; given the conjectural status of Remark 4.14, the abstract overstates the proved content. Suggest rephrasing to indicate that the basis is obtained for Hom-spaces of the oriented Temperley-Lieb category.
  2. [§2.2.2, Theorem 2.18] The proof of local confluence is a hand-check of critical pairs described in figures. The argument is plausible, but the figures for the added rules (5) and (6) are not fully detailed for all index ranges; in particular, the statement of rule (5) uses indices k ∈ [2,n−2], while the preceding discussion uses k ≤ n−3. Clarify the index bounds.
  3. [§2.2.3, Algorithm 2.2] The correctness proof of the algorithm is written in a conversational style with several 'we can therefore consider' steps. For a formal proof, the recursive calls on subwords must be shown to terminate on words with fewer generators or lower lexicographic order; currently the measure is only stated informally. Also, the notation v′_1, v″_1 is used before being defined in the pseudocode.
  4. [§3.1, Definition 3.3] The oriented algebra is defined over Z[q,q−1] with generators 1_λ and e_i, but the rewriting table in §3.2 uses q as a letter and introduces 1_λ e_i 1_μ elements. The transition from the algebra presentation to the monoid-like words is not fully formal; in particular, the idempotent relations 1_λ e_i 1_λ → 0 and 1_λ 1_μ → δ_{λ,μ} 1_λ are listed as rewriting rules but their termination is not discussed.
  5. [§4.4, Theorem 4.15] The confluence proof treats only one critical pair, saying the other is orientation-symmetric. While plausible, the proof would be stronger if it explicitly listed both critical pairs and their confluence diagrams, especially because the rules (1-left/right) and (2-left/right) are drawn in Appendix A without labels that match the theorem's numbering.
  6. [References] The reference [RS14] is cited for the Jones normal form and the standard modules, but the precise statements used (Theorem 2.4 and Proposition 2.6) are not attributed to specific locations in that paper. Also, [Mal19] is cited for Newman's lemma and modulo rewriting, but no page or chapter is given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rewriting results are checked against the independent Jones normal form, and the oriented algebra link is an openly conjectural gap, not a circular reduction.

full rationale

The non-oriented derivation is self-contained against an external benchmark. Theorem 2.18 proves convergence of the rewriting system by explicit critical-pair computations and Newman's lemma; the added rules (5) and (6) are derived from the algebra relations and are not fitted parameters. Theorem 2.19 proves, by an explicit algorithm using only rewrite rules, that the normal forms coincide with the Jones normal forms of [RS14]. That comparison is an independent check, not an assumption: JNF is defined separately (Definition 2.1), and the algorithm's correctness is argued by termination and case analysis. The oriented part is explicitly incomplete rather than circular. Theorem 4.15 establishes convergence of the rewriting system on TLO(q), and the paper concludes only that normal forms form a basis of each Hom(v,w). The further identification of Hom(v,w) for w,v in L_k with the oriented Temperley-Lieb algebra TLO_{n,k}(q) is stated in Remark 4.14 as 'everything suggests...' and is not proved. The paper also flags in Section 4.2 that the presented-category construction lacks proofs and may be incomplete, and Remark 4.8 says the bifunctor property 'should be shown'. These are missing proofs or conjectures, not circular reductions: no equation is defined in terms of its conclusion, and no fitted input is relabeled as a prediction. There are no load-bearing self-citations. The citations used for the category-algebra identifications ([Abr09], [Bow+24]) are external, and the rewriting theory is cited to [Mal19] with proofs referenced rather than reproduced. The only mild concern is the unproved bridge from Hom-space bases to an algebra basis in the oriented setting, which is a completeness/correctness issue and does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on two category-algebra bridges (one cited, one conjectural), an admittedly unpolished definition of presented monoidal category, the standard rewriting toolkit from [Mal19], and the standard planar-diagram formalism. δ and q are structural parameters of the algebra presentations, not fitted numbers; the paper contains no data fitting of any kind. Invented entities: none. The genuinely fragile inputs are the unproven identifications and the incomplete confluence checks.

assumptions (6)
  • domain assumption End_TL(δ)(n) is isomorphic to TL_n(δ)
    Proposition 4.12: proof is 'The result is stated in [Abr09]'. The non-oriented basis conclusion for Hom spaces depends on it.
  • ad hoc to paper Morphism spaces of the oriented TL category (words in L_k) are isomorphic to TLOn,k(q)
    Remark 4.14: 'everything suggests' the isomorphism; it is explicitly a conjecture and carries the oriented basis claim.
  • ad hoc to paper The presented strict monoidal category construction (Definition 4.7) is well-defined
    Author: proofs 'haven't actually done', construction 'possibly... isn't complete'; Remark 4.8 says the bifunctor property 'should be shown'.
  • domain assumption Oriented Temperley-Lieb presentation and basis from [Bow+24]
    Section 3.1: 'This algebra also admits a presentation (which I won't justify) and I refer to the article [Bow+24]'.
  • standard math Newman's lemma, Knuth-Bendix completion, modulo-rewriting confluence criteria
    Section 2.2.1 and the proof of Theorem 4.15 rely on these results from [Mal19].
  • standard math Planar diagrams up to isotopy with stacking product are well-defined
    Section 1.1: the author introduces the isotopy quotient informally ('I won't go into detail') and builds the diagrammatic algebra on it.

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Pith. "Pith review of Search for a basis of the Temperley-Lieb algebra, using rewriting systems." pith.science (2026). https://pith.science/paper/7BTTRJDH

@misc{pith2026250819360,
  author       = {Pith},
  title        = {Pith review of: Search for a basis of the Temperley-Lieb algebra, using rewriting systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BTTRJDH}},
  note         = {Machine review of arXiv:2508.19360}
}
read the original abstract

We begin by defining Temperley-Lieb algebra, in two different ways: as a presented algebra or as a diagrammatic algebra. Next, we look for a basis algorithmically, using rewriting theory. Finally, we introduce a generalization of the Temperley-Lieb algebra, which is an oriented version of the previous one. This pushes us to employ a more efficient tool, category theory, to use rewriting to easily obtain a basis for the algebra.

Figures

Figures reproduced from arXiv: 2508.19360 by the authors.

Figure 1
Figure 1. 4-diagram We identify the n-diagrams to within isotopy, i.e. we consider the quotient set Dn of the n-diagrams by the isotopy equivalence relation. I won’t go into detail on the notion of isotopy, my aim being just to have an idea to formalize the fact that two diagrams will be considered identical in our eyes if we can continuously deform, and without changing the anchor points, one to arrive at the other. Definiti… view at source ↗
Figure 2
Figure 2. The multiplication d1d2 on the left results in the diagram (here d1) with a straight link, a cup and a cap, multiplied by the scalar δ. I started by counting the diagrams, for n = 1, 2, 3 . . . , but I couldn’t see a formula for the general case. The article [RS14] bijects diagrams with what I’m going to call bridges, which consist of the same thing as diagrams but on a line: we connect 2 by 2, without crossing, 2n … view at source ↗
Figure 3
Figure 3. From a 4-diagram to an 8-bridge Theorem 1.4. Let un be the number of n-diagrams, u0 = 1 and un = Xn−1 k ukun−1−k . Proof. We can see u0 = 1 as "there’s only one diagram with 0 points, it’s the empty rectangle". I’ll count the number of bridges, noting that a bridge can only contain an even number of points so that each vertex is connected. To build a bridge with 2n points, I first construct an arc starting from the … view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: To the left of the k = 6 endpoint, I can build a 2 × 2-point bridge and 2 × 1-point bridge to the right. Remark 1.5. At this stage, I don’t have a general formula for un (depending only on n), but this property fully determines (by recurrence) a sequence. So I’ve decid…
Figure 5
Figure 5. Figure 5: Respectively 14, E1, E2, E3 These diagrams show three interesting relationships : ∀i, j ∈ [[1, n − 1]] • E 2 i = δEi (already seen in figure 2) • If |i − j| > 1, EiEj = EjEi (see figure 6) • EiEi±1Ei = Ei (see figure 7) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 = [PITH_FULL_IMA…
Figure 6
Figure 6. Figure 6: E3E1 = E1E3 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: E1E2E1 = E1 Definition 1.6. Let S be a set, and let S n = S×· · ·×S denote the set of n-uplets of elements of S. We define S ∗ = [ n∈N S n the set of finite sequences of elements of S. A finite sequence of elements of S is called word, and s1s2 . . . sn := (s1, s2, . .…
Figure 8
Figure 8. Figure 8: Non-commutative diagram 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 → 0 0 1 1 2 2 3 3 4 4 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: Image of E1 by the bijection Dn → ponts(2n) → CCSD(n) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Image of e1 by the bijection Mn/∼ → CCSD(n) 2 Basis search Now that we’ve defined the Temperley-Lieb Algebra, several questions arise. Ridout and Saint￾Aubin wonder (in [RS14]) whether there is a representation of the algebra and what it is, while Abramsky studies the…
Figure 11
Figure 11. Figure 11: E3E2E1 13 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 13
Figure 13. Figure 13: Confluence of critical pairs I’ve shown local confluence for all critical pairs except the overlap of rules (3) and (4). This is in fact the only case of non-confluence. eiei−1eiei−2 eiei−2 eiei−1ei−2ei ei−2ei (3−) (4) (4) seule possibilité (5) [PITH_FULL_IMAGE:figur…
Figure 14
Figure 14. Figure 14: Overlapping rules (3) et (4) Two different, but still equal, normal forms appear in the monoid. To remedy this, we can apply Knuth-Bendix procedure, i.e., add a rewriting rule, symbolized by the red arrow, and study the new critical pairs, with the aim of making the s…
Figure 15
Figure 15. Figure 15: Confluence of new critical pairs By repeating the method, I add a finite number of rewriting rules. In fact, the pathological case occurs for eiei−1 . . . ei−keiei−k−1, but to add the letter ei−k−1, it must exist and therefore i − k − 1 ≥ 1 ⇔ k ≤ i − 2 ≤ n − 3. I thus…
Figure 16
Figure 16. Figure 16: Exemple : visualisation du mot e1e3δe1 I’ve listed the visualizations of the rewriting rules in Appendix A. Before giving the algo￾rithm, I’d like to point out the important points of visualizing a Jones normal form. Indeed, a Jones normal form is visually a sequence …
Figure 17
Figure 17. Figure 17: Example: visualizing a Jones normal form [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: 4-oriented diagram 3.1 Presentation of algebra In this subsection, I briefly review some of the results that will be useful to me. Definition 3.1. The set SW is defined as the set of transpositions si = (i, i+1) that generate Sn, the symmetric group. For k ∈ [[1, n − …
Figure 19
Figure 19. Figure 19: Local confluence modulo [PITH_FULL_IMAGE:figures/full_fig_p031_19.png]
Figure 20
Figure 20. Figure 20: Equivalent version of local modulo confluence [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Local modulo confluence diagram of a critical pair [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: Rule (1) : eiδ → δei 0 1 2 3 4 5 6 δ e1 . . . ei → 0 1 2 3 4 5 6 δ e1 . . . ei [PITH_FULL_IMAGE:figures/full_fig_p034_22.png]
Figure 23
Figure 23. Figure 23: Rule (2) : e 2 i → δei 0 1 2 3 4 5 6 . . . . . . ei ei+1 → 0 1 2 3 4 5 6 . . . . . . ei ei+1 [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: Rule (3+) : eiei+1ei → ei 34 [PITH_FULL_IMAGE:figures/full_fig_p034_24.png]
Figure 25
Figure 25. Figure 25: Rule (3−) : eiei−1ei → ei 0 1 2 3 4 5 6 . . . ej . . . ei → 0 1 2 3 4 5 6 . . . ej . . . ei ≥ 2 [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]
Figure 26
Figure 26. Figure 26: Rule (4) : eiej → ejei si j < i − 1 0 1 2 3 4 5 6 ei−k . . . ei−2 ei−1 ei → 0 1 2 3 4 5 6 ei−k . . . ei−2 ei−1 ei [PITH_FULL_IMAGE:figures/full_fig_p035_26.png]
Figure 27
Figure 27. Figure 27: Rule (5) : eiei−1ei−2 . . . ei−kei → ei−2 . . . ei−kei 0 1 2 3 4 5 6 ei ei+1 ei+2 . . . ei+k → 0 1 2 3 4 5 6 ei ei+1 ei+2 . . . ei+k [PITH_FULL_IMAGE:figures/full_fig_p035_27.png]
Figure 28
Figure 28. Figure 28: Rule (6) : eiei+k . . . ei+2ei+1ei → eiei+k . . . ei+2 35 [PITH_FULL_IMAGE:figures/full_fig_p035_28.png]

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Works this paper leans on

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