Minimum-degree transformation representations of diagram monoids

Abstract

Abstract groups can always be faithfully represented as permutation groups.  For monoids, one needs transformations (self-maps).  This leads to a natural combinatorial problem:  finding the minimum number of points on which your monoid (or group) acts faithfully.  This parameter/invariant is called the `degree' of the monoid.

This talk reports on recent joint work with Reinis Cirpons and James Mitchell (Univ St Andrews), in which we obtain formulae for the 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 $\mathcal P_n$ 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.

If time permits, I'll briefly mention some more recent joint work with Marianne Johnson and Mark Kambites (Univ Manchester) on matrix representations of diagram monoids.