Cluster Algebras and Lusztig's Semicanonical Basis

13 June - 17 June 2011
University of Oregon
Eugene, OR

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 David Speyer.

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 Sets

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.