Representation gaps of rigid planar diagram monoids

TL;DR

This paper introduces non-pivotal variants of certain diagram monoids, analyzes their representation sizes, and evaluates their cryptographic suitability, concluding they are generally less suitable than their pivotal counterparts.

Contribution

It defines non-pivotal analogs of key diagram monoids and compares their representation gaps to assess cryptographic potential.

Findings

Non-pivotal monoids have smaller simple representations.

They are less suitable for cryptography than pivotal monoids.

Bounds for representation sizes are established.

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.