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arXiv · 2607.13445

Involutions of (twisted) diagram monoids

James East·P. A. Azeef Muhammed
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Abstract

We classify the involutions of all of the most well-studied diagram monoids -- namely the partition, planar partition, partial Brauer, Motzkin, Brauer and Temperley--Lieb monoids -- and characterise those that give rise to star-regular or regular star-monoid structures. We then complete the same program for the associated twisted diagram monoids, with respect to both the canonical float-counting twisting, and the recently-discovered rank-based twisting. This necessitates developing a general theory of involutions of twisted products. Some of our results were quite unexpected. For example, a Brauer monoid is star-regular with respect to many of its involutions, but only a regular star-monoid for one of them. We will also see that twisted diagram monoids over the integers are always star-regular, thereby providing new and very natural examples of star-regular monoids. Along the way, we also obtain (by necessity) a number of results of independent interest; specifically, we classify the automorphisms of the Motzkin and Temperley--Lieb monoids (and hence also of the planar partition monoids), and we show that all of our diagram monoids generically have trivial centre.

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James East, P. A. Azeef Muhammed. 2026-07-15. Involutions of (twisted) diagram monoids. https://arxiv.org/abs/2607.13445

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