Representation gaps of rigid planar diagram monoids

TL;DR

This work develops non-pivotal, rigid analogs of classical planar diagram monoids (rigid Temperley–Lieb, Motzkin, and planar rook) and analyzes their cryptographic viability via representation-theoretic invariants. By defining RepGap and gap ratio and applying cell theory together with explicit combinatorial counts, the authors compare non-pivotal monoids to their pivotal counterparts, showing that rigidity without pivotality generally yields smaller simple representations and less favorable gap behavior. The study combines categorical structure, Green’s relations, and explicit combinatorial computations (including truncations and GAP-based calculations) to derive asymptotics for gaps and ratios, concluding that pivotal planar diagram monoids remain preferable for cryptographic applications. Among the non-pivotal families, rpRo_n^T is semisimple with relatively strong decay in gap ratio, while rMo_n^T and rTL_n^T exhibit less secure profiles, guiding future cryptographic design toward pivotal constructions.

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 (12)

• 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 $rTL_n$ is truncated.
• Figure 2: Left: the cell structure of $rTL_{5}$, produced with the Semigroups package in GAP (adjusted to match our conventions): every block represents a $J$-cell, and every shaded box an $\mathcal{H}$-cell containing an idempotent. Right: the cell structure of $rTL_3$, with each $\mathcal{H}$-cell written explicitly.
• 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.
• Figure 4: Left: The cell structure for $rMo_3$. Right: explicit diagrams representing the single elements of the four $J$-cells of size $1$ for $k=3$.
• 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.

Theorems & Definitions (90)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Lemma 2
• Definition 1
• Remark 5
• Lemma 3
• Theorem 1