University of Sydney Algebra Seminar

James East

Friday 16 October, 12-1pm, in Carslaw 275

Minimum-degree transformation representations of diagram monoids

Cayley's Theorem states that any finite monoid can be faithfully represented as a semigroup of transformations (self-maps) of a finite set. The minimum size of such a set is the `(minimum transformation) degree' of the monoid. I’ll report on joint work with Reinis Cirpons (Inria, Nantes) 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 \(P_n\) has degree \(1 + (B(n+2) - B(n+1) + B(n)) / 2\) for \(n\geq 2\), 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 discuss some subsequent joint work with Mark Kambites and Marianne Johnson (both at Univ Manchester) on relational/matrix representations of diagram categories.