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Representation gaps of rigid planar diagram monoids

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Pith's one-line read This paper claims that rigid, non-pivotal versions of the Temperley–Lieb, Motzkin, and planar rook monoids have representation gaps exponentially smaller than their pivotal counterparts, making them generally worse for cryptographic…

desk verdict New rigid diagram monoids with a solid TL/planar-rook analysis, but the Motzkin comparison has a load-bearing inconsistency and an unavailable key reference. read the letter →

arxiv 2505.05846 v1 pith:UM2PPUMB submitted 2025-05-09 math.RT cs.CRmath.GRmath.QA

Willow Stewart , Daniel Tubbenhauer This is my paper · ORCID
classification math.RTcs.CRmath.GRmath.QA MSC 18D1020M3005A1694A60
keywords representationgapdiagrammonoidsTemperley–LiebmonoidMotzkinplanarrookrigidmonoidalcategoriesnon-pivotalcryptography
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 asks whether rigid-but-not-pivotal versions of familiar planar diagram monoids are better or worse as cryptographic building blocks, where 'better' means having large representation gaps: large minimal dimensions of nontrivial simple representations, which resist linear-algebra attacks. The authors construct non-pivotal analogs of the Temperley–Lieb, Motzkin, and planar rook monoids and compute asymptotic bounds for their RepGaps and gap ratios. Their central claim is that, contrary to the intuition that a more complicated category should yield a harder algebra, all three non-pivotal monoids have representation gaps that are exponentially worse than their pivotal counterparts. If correct, the conclusion matters for protocol design: among these families, pivotal planar diagram monoids remain the safer candidates, while the non-pivotal ones are comparatively easy to attack by linear methods.

What carries the argument

The machinery is cell theory for finite monoids, using Green's relations to decompose diagrams by the number of through strands. For the rigid Temperley–Lieb monoid $\mathrm{rTL}_n$, every diagram factors as $\tau \circ 1_k \circ \beta$ with $k$ even through strands, giving right cells of size $\binom{n-1}{(2n-k)/2}$ and explicit simple-representation dimensions. For the rigid Motzkin monoid, through-strand sequences form a poset and right-cell sizes are computed via Catalan $k$-fold convolutions, simplified with a hypergeometric Chu–Vandermonde identity. For the rigid planar rook monoid, the same sequence combinatorics gives binomial cell sizes and proves the monoid is semisimple over any field. Truncation, by taking Rees quotients that keep only the dominant through-strand interval, is used so that the smallest nontrivial simple representation determines the gap.

What would settle it

Compute the smallest nontrivial simple representation of the pivotal Motzkin monoid $\mathrm{Mo}_n$ for $n = 10$ through $20$ by direct Gram-matrix rank calculation: if the exponential growth factor is below $9^n$, then Corollary 5C.8's comparison collapses.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that passing from a pivotal to a non-pivotal rigid planar diagram monoid lowers the exponential growth of the RepGap. For the rigid Temperley–Lieb monoid, the gap lies between $\Omega(n^{-1/2} 2^n)$ and $O(n^{-1/2} 2^n)$, whereas the pivotal TL monoid has gap between $\Omega(n^{-5/2} 4^n)$ and $O(n^{-3/2} 4^n)$. For the rigid Motzkin monoid, the semisimple gap lies between $\Omega(n^{-3/2} 4^n)$ and $O(n^{-1} 4^n)$, while the pivotal Motzkin gap has a lower bound of $\Omega(9^n)$. For the rigid planar rook monoid, the gap is $O(n^{-1/2} 2^n)$ versus a pivotal lower bound of $\Omega(n^{-1/2} 4^n)$, and its gap ratio decays at most like $0.87^n$. These comparisons support the paper's conclusion that the rigid non-pivotal monoids are generally worse for cryptographic purposes, despite the rigid Temperley–Lieb monoid having a slightly better gap ratio.

Load-bearing premise

The comparison depends on the unpublished lower bound that the pivotal Motzkin monoid's RepGap grows like $9^n$; if that bound is wrong or overstated, the paper's main conclusion for the Motzkin case loses its support.

Editorial extensions

If this is right

  • If the bounds are correct, a cryptosystem built on the rigid Temperley–Lieb monoid can be reduced to linear algebra of dimension at most $O(n^{-1/2} 2^n)$, while the pivotal version offers dimension at least $\Omega(n^{-5/2} 4^n)$; the rigid version is exponentially easier to attack.
  • For the Motzkin monoid, even the semisimple dimension, which is an upper bound for the true simple dimension, grows only like $4^n$ in exponential base for the rigid version, compared with a $9^n$ lower bound for the pivotal version, so the rigid version cannot offer comparable security.
  • For the planar rook monoid, the rigid gap ratio decays like $0.87^n$, meaning the gap becomes exponentially small relative to the square root of the monoid size, the weakest profile among the monoids studied.
  • The three cases together support the paper's general statement that pivotal planar diagram monoids are better suited than their non-pivotal analogs for the cryptographic applications considered.
  • The comparison uses the paper's more generous RepGap definition, which agrees with the earlier definition on the reference monoids, so the conclusion is not an artifact of a stricter gap definition.

Reading between the lines

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

  • Editorial inference: the recurring $2^n$-versus-$4^n$ pattern suggests a broader principle—breaking pivotal symmetry may halve the exponential base of the smallest simple representation, so other two-colour or oriented diagram monoids could show similar drops if constructed along the same lines.
  • One stress test is to compute RepGaps for the full untruncated monoids or for different truncation widths; the paper's conclusions are stated for the chosen truncations, and it is not yet known whether the exponential separation survives under all reasonable truncation choices.
  • Because the rigid Temperley–Lieb monoid is characteristic-free while the pivotal one changes dramatically in prime characteristic, the cryptographic ranking drawn in characteristic zero may not transfer to protocols over finite fields; positive-characteristic analysis would settle that.
  • If the unpublished $9^n$ Motzkin bound is independently verified, the same cell-size formulas could be sharpened to give explicit constants, turning the asymptotic $\Omega$ and $O$ bounds into $\Theta$ estimates.
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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 / 5 minor

Summary. The paper defines rigid, non-pivotal analogues of the Temperley–Lieb, Motzkin, and planar rook monoids, computes bounds on the sizes of their simple (or cell) representations, and compares the resulting representation gaps and gap ratios with the pivotal versions. It concludes that the non-pivotal monoids are generally worse for cryptographic purposes.

Significance. If the technical claims are correct, the paper provides the first systematic comparison of pivotal versus non-pivotal planar diagram monoids from the point of view of representation gaps, with explicit cell-size formulas and concrete asymptotic bounds. The rTL and rpRo legs rest on self-contained derivations, and the paper ships reproducible code on GitHub for the computer-assisted parts. However, the Motzkin comparison depends on an unpublished thesis, and the headline claims about the rigid Motzkin monoid are stated for the RepGap while only semisimple bounds are proved.

major comments (4)
  1. [§3A, Theorem 3A.3] The two displayed bounds for Gap_K(MoT_n), namely f(n)·9^n ≤ Gap_K(MoT_n) ≤ 2^{-3/2} n^{-3/2} 4^n, are mutually incompatible for large n because (9/4)^n grows exponentially. As written, at least one of the bounds is wrong or mis-stated; this directly affects Corollary 5C.8 and the Motzkin row of the table in the introduction.
  2. [§3A, proof of Theorem 3A.3] The pivotal Motzkin lower bound is cited to [Ar25], an unpublished honours thesis with no public text. This bound is load-bearing for the paper's central comparison, so the authors should either give a self-contained proof of the 9^n exponential growth or make the thesis available and ensure the quoted statement is correct, especially in light of the inconsistency noted above.
  3. [§1B and §5C] The abstract and Table 1 report Gap^{1/n}→4 and Ratio^{1/n}→1 for the rigid Motzkin monoid, but Theorem 3B.6 and Theorem 5C.6 only establish bounds on the semisimple gap ssGap. Since Gap ≤ ssGap by Lemma 2A.10, these bounds do not determine the exponential growth of the actual RepGap; the claims must be rephrased as semisimple-gap statements or supplemented with genuine RepGap bounds.
  4. [§4C, Proposition 4C.1] The proof that the right cell representations of rTL_n are simple is sketched and contains an internal inconsistency: the text first states that wx ∉ W for a suitable x, but the displayed computation proves xw ∉ W. The proof also assumes without justification that a nontrivial invariant subspace W contains an element with at least one cap. Because this proposition underpins the rTL gap bounds, a complete and correct proof is needed.
minor comments (5)
  1. [§5C, Corollary 5C.8] The notation '∋ Ratio_K MoT_n' is non-standard and should be replaced by an explicit statement such as 'Ratio_K(MoT_n) ∈ [Ω((1−ε)^n), O((1+ε)^n)]'.
  2. [§6B, Proposition 6B.1] In the displayed formula for |rpRo_n|, the factor 'binom{n+j}{2}^2' should read 'binom{n+j}{k}^2'.
  3. [§6C, Theorem 6C.4] The upper bound for the gap ratio is typeset as '23n/233n/45−5n/4'; this should be 2^{3n/2}3^{3n/4}5^{-5n/4}.
  4. [Remark 3B.5] There is a typo: 'the difference wll not play a role' should be 'will not play a role'.
  5. [References] The bibliography entry [CG23] lists the authors as 'S. Doty and A. Giaquinto', but the citation key begins with 'C'; this mismatch should be fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-pivotal RepGap bounds are derived from explicit cell counts and Stirling asymptotics, while the pivotal benchmarks are external prior results; no prediction is equivalent to its input by construction.

full rationale

The non-pivotal bounds in Theorems 4C.3, 5C.6, and 6C.4 are derived from independently computed cell sizes (e.g. Proposition 4B.6: |R_k^n| = binom(n-1,(2n-k)/2), and Proposition 5B.13 for the Motzkin cells), followed by Stirling asymptotics and explicit Catalan/hypergeometric identities. These quantities are not fitted to the pivotal comparison and do not presuppose the paper's conclusion. The pivotal benchmarks are quoted from prior published work [KST24] for TL and pRo, and from the unpublished honours thesis [Ar25] for the Motzkin 9^n lower bound; even though [KST24] shares an author, it is external published support rather than a self-imported uniqueness principle, and the non-pivotal computations do not reduce to those citations. The [Ar25] dependence is a verifiability risk, not a circular step, because the paper does not define its own Motzkin gap in terms of that thesis or fit anything to it. One genuine non-circular correctness concern: Theorem 3A.3 as printed asserts both f(n) 9^n <= Gap_K(MoT_n) and Gap_K(MoT_n) <= 2^{-3/2} n^{-3/2} 4^n, which are incompatible for large n since (9/4)^n grows exponentially; this suggests a typographical or attribution error in the theorem, but it is an internal consistency issue rather than a circular derivation. Since no load-bearing step exhibits a definitional reduction, a fitted input renamed as a prediction, or a self-citation chain that forces the result, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's own derivations use standard cell theory and asymptotics. The main non-standard input is [LS25] for Catalan convolutions and [Ar25] for the pivotal Motzkin lower bound. No numerical constants are fitted to data; truncation parameters are inherited from [KST24]. The semisimple-dimension upper bound (Lemma 2A.10) is the key tool for rMo, and the authors acknowledge it overestimates the true representation gap (Remark 5C.1).

free parameters (1)
  • truncation cutoff width = 2√(2n) for rMo; n/2 ± √(2n) for rpRo; n+1 ± √(2n) for rTL
    Chosen by hand to remove small cells while retaining the bulk of the monoid, inherited from [KST24] for pivotal cases. The reported gap ratio is sensitive to this choice; for rMo the full-monoid ratio is worse than the truncated ratio.
assumptions (5)
  • standard math Clifford-Munn-Ponizovskii theorem (Theorem 2A.9): simple M-representations correspond 1:1 to idempotent J-cells for monoids with trivial H-cells.
    Cited from [KST24] and used to identify simple representations with cell representations throughout.
  • standard math Lemma 2A.10: dim(L_J) <= |L| for any left cell L in a J-cell.
    Used to bound the RepGap by the semisimple dimension; central for the rMo analysis.
  • standard math Catalan k-fold convolution formula ([LS25, Proposition 1.2]) for sums of products of Catalan numbers.
    Used to simplify the cell-size sums in Proposition 5B.13.
  • standard math Stirling approximation and Mathematica asymptotic expansions for binomial coefficients and Gamma functions.
    Used throughout for asymptotics; results are stated without full derivations, relying on Mathematica computations.
  • standard math Motzkin number asymptotic |Mo_n| ~ (3^{4n+3/2})/(16√π n^{3/2}) from [Ko12].
    Used for the pivotal Motzkin gap ratio in Theorem 3A.3.

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Pith. "Pith review of Representation gaps of rigid planar diagram monoids." pith.science (2026). https://pith.science/paper/UM2PPUMB

@misc{pith2026250505846,
  author       = {Pith},
  title        = {Pith review of: Representation gaps of rigid planar diagram monoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UM2PPUMB}},
  note         = {Machine review of arXiv:2505.05846}
}
read the original abstract

We define non-pivotal analogs of the Temperley-Lieb, Motzkin, and planar rook monoids, and compute bounds for the sizes of their nontrivial simple representations. From this, we assess the two types of monoids in their relative suitability for use in cryptography by comparing their representation gaps and gap ratios. We conclude that the non-pivotal monoids are generally worse for cryptographic purposes.

Figures

Figures reproduced from arXiv: 2505.05846 by the authors.

Figure 1
Figure 1. Left: comparison of the RepGaps in Log10 scale up to n = 1000; Right: comparison of the gap ratios in Log10 scale at n = 100, with vertical lines indicating where rT Ln is truncated. to determine that pivotal planar diagram monoids are generally better for cryptographic purposes than non-pivotal planar diagram monoids. The results related to the symmetric monoids in [KST24, Section 5] are much less precise than thos… view at source ↗
Figure 2
Figure 2. Left: the cell structure of rT L5, produced with the Semigroups package in GAP (adjusted to match our conventions): every block represents a J-cell, and every shaded box an H-cell containing an idempotent. Right: the cell structure of rT L3, with each H-cell written explicitly. Now, we calculate the sizes of the left and right cells. Proposition 4B.6. For a fixed number of through strands k, we have: |Rk n | =  n −… view at source ↗
Figure 3
Figure 3. Left: The dimension of the simple representations over k, for n = 100, with vertical lines marking the truncation end points. Right: A Log10 version of the same graph. Now we can find the RepGap and gap ratio, and compare the results with those for T Ln. Theorem 4C.3. We have the following inequalities for the RepGaps and gap ratios of rT LT n 2 −1/2π −1/2 e −1− 1 3n · n −1/2 · 2 n ≤ GapK rT LT n ≤ 2 −1/2π −1/2 · n … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Left: The cell structure for rMo3. Right: explicit diagrams representing the single elements of the four J-cells of size 1 for k = 3. If we pair the first 2 with the ith 1 following that 2, we have 1 2...1 2 of length 2i − 2 under the cap, and 1 2...1 2 of length 2l − …
Figure 5
Figure 5. Figure 5: Left: the semisimple dimension over k, for n = 100, with a vertical line showing the rightmost truncation endpoint. Right: A Log10 version of the same graph. We again denote this truncated monoid by rMoT n . Theorem 5C.6. We have the following. π −1/2 e − 1 n · n −3/2 …
Figure 6
Figure 6. Figure 6: Left: the bulk of the monoid rMon, with the rightmost truncation end￾point marked by a vertical line, showing the truncation is much smaller than the monoid size. Right: A Log10 version of the same graph. This also reveals that the asymptotic of the nth root of the rat…
Figure 7
Figure 7. Figure 7: Visual comparison of the RepGaps between the non-pivotal and pivotal Motzkin monoid, on a Log10 plot [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: Left: The cell structure for rpRo3. Right: explicit diagrams representing the single elements of the four J-cells of size 1 for k = 3. 6C. RepGap and gap ratio of rpRon. The key difference between rpRon rMon is that rpRon is semisimple, and this follows from the follow…
Figure 9
Figure 9. Figure 9: Left: the dimension of the simple representations over k, for n = 200, with vertical lines marking the truncation end points. Right: A Log10 version of the same graph. Theorem 6C.4. The asymptotics of the RepGap and gap ratio of rpRoT n have the following inequal￾ities…
Figure 10
Figure 10. Figure 10: Left: the bulk of the monoid rpRon, with truncation endpoints marked by vertical lines, showing the truncation is much smaller than the monoid size. Right: A Log10 version of the same graph. that it is monotone increasing in n, and calculating the result for n = 500 i…
Figure 11
Figure 11. Figure 11: Visual comparison of the RepGaps between the non-pivotal and pivotal planar rook monoid, on a Log10 plot [PITH_FULL_IMAGE:figures/full_fig_p036_11.png]
Figure 12
Figure 12. Figure 12: Visual comparison of the gap ratios between the non-pivotal and pivotal planar rook monoid, on a Log10 plot. References [An19] H.H. Andersen. Simple modules for Temperley–Lieb algebras and related algebras. J. Algebra, 520:276–308, 2019. URL: https://arxiv.org/abs/170…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.