Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleEnumerating (Garside) lattices
PresenterVolker Gebhardt (University of Western Sydney)
Author(s)Volker Gebhardt, Stephen Tawn
Sessionn/a
Time
Abstract


Generating catalogues of examples that are in some sense complete has proved to
be an important step towards understanding  algebraic concepts, and it often is
one of the key steps towards a successful classification and thus a complete
theory.

One type of algebraic structure I am particularly interested in are so-called
\emph{Garside monoids}:
This is a class of infinite monoids that admit a ``nice'' (and effectively
computable) normal form for their elements.
The notion of Garside monoids captures and unifies many important examples, for
instance all Artin groups of spherical type, free groups, free abelian groups,
as well as (at least some) mapping class groups, complex reflection groups, and
affine Artin groups.%
\bigskip

Garside monoids can be defined by a finite lattice (in the meaning of
combinatorics), together with a labelling of the edges that satisfies certain
conditions.
Thus, one can catalogue the Garside monoids up to some size threshold by:
\begin{enumerate}
 \item[(a)] Constructing all lattices on at most $n$ points.
 \item[(b)] Constructing all suitable edge labellings of a given finite
lattice.
\end{enumerate}
\bigskip

It turns out that (b) isn't too bad.  (In the unlikely case that there is time,
I'll mention the key ideas.)

Unfortunately, (a) is: even the \emph{number} of lattices on $n$ points is
currently only known for $n\le 19$ (OEIS: A006966).
\bigskip

In my talk, I'll focus on explaining how to combine group theory, combinatorics
and some clever ideas from computing to enumerate unlabelled lattices more
effectively.