Minimum transformation representations of diagram monoids

TL;DR

This paper derives explicit formulas for the minimum transformation degrees of key finite diagram monoids, providing new insights into their algebraic representations and explicit faithful models.

Contribution

It introduces formulas for the minimum transformation degrees of various diagram monoids, including partition, Brauer, Temperley–Lieb, and Motzkin monoids, with explicit constructions.

Findings

Partition monoid degree formula involving Bell numbers

Explicit faithful representations constructed for these monoids

Results applicable to understanding algebraic actions on projections

Abstract

We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid $P_n$ has degree $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq2$, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.