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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.
After the seminar we will take the speaker to lunch. See the Algebra Seminar web page for information about other seminars in the series. Anthony Henderson [email protected]. |