Cluster Algebras and Lusztig's Semicanonical Basis
13 June - 17 June 2011
University of Oregon
Let G be a simply connected semisimple algebraic group, B a Borel subgroup, and N its nilpotent radical.
In 1990, Lusztig constructed a really awesome basis (the canonical
basis) for the universal enveloping algebra U(n) using perverse sheaves
on a variety that parameterizes representations of a quiver.
In 2000, he constructed a slightly different, equally awesome basis
(the semicanonical basis) using constructible functions
on a closely related variety.
Dually, we obtain two bases for the ring of functions on G/N.
In 2001, Fomin and Zelevinsky
introduced the notion of a cluster algebra, with the ring of functions on SL(3)/N being one of the first examples.
A cluster algebra has certain distinguished elements called cluster
variables and certain monomials in the cluster variables are singled
out and labeled as cluster monomials. In the case of
SL(3)/N, the cluster monomials coincide with both the dual canonical
and dual semicanonical basis.
More generally, Geiss, Leclerc, and Schröer showed in 2007
that the cluster monomials are contained in the dual semicanonical basis;
it is conjectured that they lie in the dual canonical basis, as well.
The goal of the workshop will be to understand these statements.
My advisor taught me that a good way to learn math is to read
lots of ICM addresses. Here are two on the subject of this conference,
by Lusztig from 1990 (Section 19) and Leclerc from 2010 (Sections 1-6).
This 2008 survey article by Geiss, Leclerc, and Schröer is also a good reference.
The workshop will be aimed at graduate students and postdocs, with most
of the talks given by the participants. We do not expect any of the
participants to be experts in all of the subjects that are represented
in this workshop. Rather, we hope to bring together participants
with diverse backgrounds, and to weave these backgrounds together into a coherent
picture through a combination of lectures and informal discussion sessions.
The workshop will be led by
Schedule and References
A schedule of talks is available here.
I have also compiled a list of references
on canonical and semicanonical bases, to be used for this workshop
and for a class that I'm teaching this spring,
along with historical notes and links to electronic versions of all the papers.
Daniel Moseley has been livetexing the workshop; his notes
are available here.
Problem Set 1
Problem Set 2
Problem Set 3
Here is a map that contains the motel, the lecture
hall, and a few places to eat and drink.
Here is a map to McMenamin's North Bank, where we will
have the problem sessions on Tuesday and Thursday at 8:00.
The deadline of March 1st to request funding has passed.
Other participants are still welcome if they can cover their own expenses,
subject to hotel room availability.
This workshop is part of a series of annual workshops funded
by an NSF CAREER grant. The 2010 workshop, led by Andre Henriques,
was on the subject of Operator Algebras and Conformal Field Theory.