Organizers :
Andrew Carroll,
Steven Sam,
Salvatore Stella,
Peter Tingley Time: Thursday 6pm-7:30pm Place: We will be alternating in between MIT and Neu. Room @ MIT: 2-132 (MIT math department, building 2 on
this map). Room @ Neu: 265 Ryder Hall (RY): it is building 24 on
this map. Next meeting: Dec 8st @ NEU
News
10/03: Again a change of room at Northeastern (hopefully the last). We
now are at 265 RY; same building different floor. 09/28: New room at Norteastern: 156 RY; same building same floor
different room
Plan
The main idea behind this seminar is to introduce cluster algebras with a view
towards understanding cluster categories and tilting theory.
At the moment the plan is pretty flexible and it will be adapted to any
input the audience will provide at the first meeting.
The first few lectures will be devoted to fixing a common background.
Cluster algebras will be defined through working examples. Since the
cluster category is constructed as a category of representations of a
quiver algebra, we will also have some introductory lectures on quiver
algebras and the related notions. Once the background has been
established, we'll explore some other aspects of cluster algebras,
perhaps including recent work on cluster algebras coming from surfaces,
and Nakajima's work on quiver varieties
The cluster category is a beautiful example of a category with operations that
mimic those in cluster algebras. Such relationships are a subject of much
current interest, under the vague heading of "categorification." Depending on
how the semester develops, we are planning to at some point transition to more
general categorification topics, and specifically the categorification of
quantum sl(2) given by Aaron Lauda.
Ideally talks should be given by students not familiar with the topics with the
assistance of the organizer and fellow students more familiar with the subject.
Therefore we strongly encourage to participate to the first seminar to volunteer
and pick up a talk.
9/15 @ NEU Andrew Carrol: An introduction to quivers and Auslander-Reiten
theory
Quiver representation theory arose from the systematic study of a wide
variety of linear algebra-type problems, and has grown into an interesting and
fertile ground for studying. They have been used to solve the Deligne-Simpson
problem in differential equations, prove conjectures in the realm of cluster
algebras, and their study proves to yield beautiful combinatorics. This talk
will be an introduction to quivers and their representations. I will point out
some benchmarks in the development of the theory, and discuss Auslander-Reiten
theory which is a strong tool for organizing the category of representations of
a quiver.
9/22 @ MIT Peter Tingley: Canonical bases, dual canonical bases, and
positivity
Fix a complex simple Lie group G and let g be its Lie algebra. I will
give an introduction the canonical bases for U^-(g) (the lower
triangular part of the universal enveloping algebra), and the dual
canonical basis for the function algebra of N (the lower triangular
unipotent subgroup of G). This will be done via Kashiwara's crystal
bases, which I will also explain. I'll then discuss some positivity
properties of these bases. The development of cluster algebras was
largely motivated by this type of positivity. I will discuss all of
this using the example sl(3), which should be accessible without much
background. The structure in this case is perhaps over simplified, but
there is enough visible to serve as motivation.
9/29 @ NEU
Nan Li: Introduction to cluster algebras
Cluster algebra is a class of commutative ring, introduced in 2000 by Fomin
and Zelevinsky, originally to study Lusztig's dual canonical basis and total
positivity. After that, connections have been made to many other fields,
including coordinate rings of Grassmannians, quiver representations,
Teichmuller theory, invariant theory, tropical calculus, combinatorics, etc.
In this introduction, we will start from the example of pentagon recurrence.
Then give the definition of cluster algebra and state the Laurent phenomenon
and the positivity conjecture. We will then explain the example of
triangulation of an n-gon (coordinate ring of G(2,n)). Finally, the
connection with quiver mutations will be mentioned. This introduction is
based on Lauren Williams' 1st lecture in a summer shool on cluster algebras
at MSRI this August.
10/6 @ NEU
Gordana Todorov: Cluster categories
Cluster categories were introduced in order to give
categorical interpretation to the combinatorics of the cluster algebras
of Fomin and Zelevinsky.
To an acyclic quiver Q, and a field k, we associate path algebra kQ,
which is hereditary, i.e. all modules have projective dimension at most
1. The derived category of bounded complexes is quite easy to describe
in this case, and it is a triangulated category. The shift functor,
composed with the Auslander- Reiten translation defines a triangulated
functor, which has the property that the orbit category is also
triangulated. The cluster category associated to the quiver Q is defined
to be that orbit category.
Cluster tilting objects in the cluster category are defined to be
objects with no selfextensions and with n indecomposable non-isomorphic
summands, where n is the number of vertices in the quiver Q. Cluster
tilting objects have the property that each indecomposable summand of a
cluster tilting object can be replaced in exactly one way by another
indecomposable object in such a way that a new cluster tilting object is
obtained. This procedure is done using approximations in the cluster
category.
At this point, I will recall the basic properties of cluster algebras
and state the precise correspondence between the indecomposable rigid
objects, cluster tilting objects, cluster tilting mutations on one side,
and cluster variables, clusters and cluster mutations on the other side.
Throughout the lecture I will be using example of the quiver A4 to
illustrate all the notions and correspondence with cluster algebras.
10/13 @ MIT
Alejandro Morales: An overview of the packages for cluster algebras and quivers
in SAGE
I will give a demo of the recent cluster and quiver packages written by Musiker
and Stump for sage, a free open source mathematics software. We will build from
the examples seen in Nan Li's talk, especially about the associahedron in
different types. The main reference for this talk is the compendium of the
packages by Musiker and Stump (http://arxiv.org/pdf/1102.4844v2).
The demo will be accessible to people with little or no experience in sage. See
http://www.sagemath.org/ and http://wiki.sagemath.org/combinat on
how to install the software and
http://www.math.umn.edu/~musiker/8680/SAGE-Cluster/ on how to install the
package.
10/20 @ NEU
Leo Petrov: Totally nonnegative Toeplitz matrices and characters of "big" groups
I will try to explain the classification of totally nonnegative Toeplitz
matrices and its connections with the representation theory of "big" groups,
namely, the infinite symmetric group and the infinite-dimensional unitary group.
Take an irreducible character of the infinite-dimensional unitary group U(\infty)
and restrict it to the finite-dimensional subgroup U(N). Then the restriction
is a reducible character of U(N). Its decomposition into irreducibles is governed by
minors of a suitable Toeplitz matrix. Thus the minors must be nonnegative.
Moreover, it turns out that the whole classification of totally nonnegative
Toeplitz matrices (obtained by Shoenberg and Edrei around 1950) is equivalent
to the classification of the irreducible characters of U(\infty) (which is independently
due to Voiculescu, 1978). The triangular totally nonnegative Toeplitz matrices
correspond to characters of the infinite symmetric group, and their classification was
obtained by Thoma in 1964 (also independently of works of Shoenberg and his
followers).
10/27 @ MIT
Barbara Bolognese: Cluster Algebras Arising from Surfaces
In the talk, I will give generalities about the
triangulation of marked Riemann surfaces with boundary, explaining how
this leads to
the construction of certain cluster algebras and certain cluster
categories. Then I will construct a formula which allows to prove the
positivity conjecture from cluster algebras from surfaces and,
eventually, I will explain how triangulations allow us to give a
combinatorial characterization of the mapping class group.
11/3 @ NEU
Steven Sam: Double Bruhat cells and total positivity
A matrix is totally positive if all of its minors are
nonnegative. One can think of this as a "type A" definition, and
Lusztig generalized the notion of total positivity to any semisimple
Lie group. Of importance in this notion are the double Bruhat cells:
intersections of two Bruhat decompositions corresponding to an opposed
pair of Borel subgroups. These cells are birationally equivalent to
affine space, and Fomin-Zelevinsky constructed functions which allow
one to test total positivity, which generalizes the notion of minors.
I'll explain this story and we will see some connections to exchange
relations.
11/10 @ MIT
Jerzy Weyman: Clusters, Quivers, Pictures
This will be an introduction to cluster tilted algebras and their connections to
semi-invariants of quivers and general decompositions of quiver representations.
I will also mentioned some open problems related to representations of quivers
that are open even in finite representation type case.
11/17 @ Neu
Ryan Kinser: Preprojective algebras
We will cover the background on preprojective algebras which
is necessary to understand a future talk on Geiss, Leclerc, and
Schroer's work relating preprojective algebras and cluster algebras.
11/24
Thanksgiving
12/1 @ MIT
Salvatore Stella: Preprojective algebras and cluster algebras
Using the module category of preprojective algebras, Geiss, Leclerc and
Schröer gave a categorification of a class of cluster algebras introduced by
Berenstein, Fomin and Zelevinsky in relation with their works on total
positivity in semisimple groups. This allowed them to prove that the cluster
monomials of those algebras belong to the dual of Lusztig's semicanonical
basis. We will give an overview of the general construction they give and
provide some examples.
12/8 @ NEU
Vinoth NandaKumar: Cluster algebras and representations of quantum affine algebras
I will give an overview of work by Hernandez and Leclerc which constructs
a monoidal categorification of cluster algebras of finite type via a
certain subcategory of representations of the quantum affine algebra. I
will describe some of the basic theory of representation of quantum affine
algebras (such as q-characters, and Drinfeldt polynomials), and show some
of the ideas behind the proof of the conjecture in Type A by looking at
some small examples in detail.
Last modified:
Tuesday, 13-Dec-2011 15:15:27 EST