Algebra Seminar: East -- Minimum-degree transformation representations of diagram monoids

James East (Western Sydney University) will be speaking in the algebra seminar.  We will
go out for lunch after the talk---all are welcome to join! 

When: 12-1pm Friday October 16 

Where: Carslaw 275 

Title: Minimum-degree transformation representations of diagram monoids 

Abstract: 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>=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.