James East (University of Sydney)

Friday 4th April, 12.05-12.55pm, Carslaw 373

Braid groups and transformation semigroups

Braid groups may be thought of as "rigidified" symmetric groups. Abstractly, they are obtained by deleting the order relations from the classical symmetric group presentation; geometrically, they arise when one considers the lines in the diagrammatic representation of a permutation to be strings which may cross in a positive or negative fashion. But what happens when we try to "rigidify" other objects? Symmetric groups may be generalised in many different ways, depending on the interests of the generaliser. Semigroup theorists consider transformation semigroups (of which the symmetric group is a pivotal example), and rigidified examples of some transformation semigroups have been explored by numerous authors, giving rise to some interesting monoids of braids (and braid-like objects). On the other hand, group theorists consider Coxeter groups (the symmetric groups being the Coxeter groups of type A). In this talk we will focus on a single type of braid monoid, the so-called partial vine monoid, to illustrate the semigroup theoretic side of the story. If time permits, we will look at some recent work on the Coxeter group side.