Last updated: April 24, 2008 23:44 EST

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International Conference on Representations of Algebras and Related Topics
April 25 - 27, 2008
Woods Hole, Massachusetts, USA

All lectures are in 104/105 Candle House
Friday, April 25
1:00 - 2:00 Registration (Swope Building first, then 104/105 Candle House)
2:00 - 2:35 Hugh Thomas Coloured quiver mutation for higher cluster categories
2:45 - 3:20 Mark Kleiner Preprojective elements of Coxeter groups
3:30 - 4:05 Shih-Wei Yang
 Combinatorics of Coxeter elements and cluster algebras of
finite type
Coffee break
4:30 - 5:30 Bill Crawley-Boevey Problems in Linear Algebra Related to Root Systems
Reception and Dinner


Saturday, April 26

 
9:30 - 10:30 Bill Crawley-Boevey Indecomposable Representations of Quivers
Coffee break
10:45 - 11:20 Mara Neusel   On the Hilbert Ideal
11:30 - 12:05 Ryan Kinser Rank functors and representation rings of quivers

Lunch break
2:00 - 2:35 Frauke Bleher Universal deformation rings related to the
symmetric group on 4 letters
2:45 - 3:20 Lars Christensen Finite Gorenstein Representation Type implies
Simple Singularity
Coffee break
3:45 - 4:20 Shiping Liu Derived-finite algebras
4:30 - 5:05 Kristin Webster Semi-invariants
5:15 - 5:50 Franco Saliola Two algebras related to finite Coxeter groups
Dinner break
8:30 - 9:30 Informal talk (tentative)
 
Sunday, April 27
9:00 - 9:35 Ralf Schiffler Cluster tilted algebras and slices
9:45 - 10:15 Charles Paquette Triangulated stable module categories
Coffee break
10:45- 11:20 Thomas Brustle Algebraic structures given by triangulations of surfaces
without punctures
11:30- 12:05 Daniel Labardini Fargoso Mutations and Representations of Quivers with Potentials
of Surface Triangulations
12:15- 12:50 Osamu Iyama Mutation of cluster tilting object and potentials

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Titles and Abstracts
Frauke Bleher, Universal deformation rings related to the symmetric group on 4 letters

Abstract: This is joint work with Ted Chinburg and Jennifer Froelich. In the eighties, Mazur, using work of Schlessinger, introduced techniques of deformation theory in the study of representations of Galois groups. In this talk we will determine the universal deformation rings of certain representations over Z/2Z of the symmetric group S4 and one of its double covers and show how these deformation rings arise in arithmetic.

Thomas Brustle, Algebraic structures given by triangulations of surfaces without punctures.

Abstract: We study the quiver with potential (Q,W) which encodes a triangulation of an oriented marked surface (compare to Daniel Labardini, who considers the more general case of a surface with punctures). We show that homotopy classes of marked curves in the surface correspond to string modules over the Jacobian algebra J(Q,W). This provides a "categorification" of those curves, and we explicitly describe the irreducible morphisms and the Auslander-Reiten translation.

Lars Christensen, Finite Gorenstein representation type implies simple singularity.

Abstract: The germs of regular functions at a point on an algebraic variety form a local ring; this makes local rings a fundamental object of study in commutative algebra.
    Representation theory of local rings is concerned with connections between the module category of a ring and the character of its singularity. When it is not possible to give an exhaustive description of the module category by classifying its indecomposable objects, one hopes to find characteristics of the ring reflected in a manageable subcategory of this category. 
    Striking examples of such connections emerged in the 1980s.  They show how finiteness conditions on the category of maximal Cohen-Macaulay modules characterize particular isolated singularities. I will discuss some recent improvements of these results; they have been obtained in collaboration with Greg Piepmeyer, Janet Striuli, and Ryo Takahashi.

Daniel Labardini Fragoso, Mutations and representations of quivers with potentials of surface triangulations.

Abstract: Recently, H.Derksen, J.Weyman and A.Zelevinsky have introduced quivers with potentials (QPs) and their mutations, thus providing a new representation-theoretic interpretation for quiver mutations originated in the theory of cluster algebras. In another recent development, S.Fomin, M.Shapiro and D.Thurston have introduced a class of quivers associated with triangulated oriented bordered surfaces with marked points. This class is invariant under quiver mutations, which correspond to geometrically defined flips on ideal triangulations. In this talk we discuss the problem of lifting these flips to the level of QPs. We also discuss possible generalizations of a recent work by I.Assem, T.Brüstle, G.Charbonneau-Jodoin and P-G.Plamondon in the context of mutations of QP-representations, with the aim of lifting the flips further to the level of representations.

Shiping Liu, Derived-finite algebras.

Abstract: TBA

Osamu Iyama, Mutation of cluster tilting object and potentials

Abstract: This is a joint work with Buan, Reiten and Smith. We show that mutation of cluster tilting object in 2-CY triangulated categories (resp. tilting modules over 3-CY algebras) is compatible with Derksen-Weyman-Zelevinsky mutation of quivers with potentials by observing change of 2-almost split sequences. We apply our results to 2-CY triangulated categories associated with Coxeter group
elements.

Ryan Kinser, Rank functors and representation rings of quivers.

Abstract: We'll discuss our recently constructed functor which takes a representation of an arbitrary (finite) quiver Q, and returns a representation of Q for which the maps over all arrows are isomorphisms.  The common dimension of the resulting vector spaces at each vertex is a numerical invariant of the representation.  Combining these functors with pulling back representations along well chosen maps of directed graphs allows us to construct other numerical invariants of representations.  These include, as the simplest cases, the dimension vector of a representation, the ranks of any composition of maps, dimensions of intersections of images, and so forth.  We call the functors giving rise to these invariants "rank functors", although in general they measure something more complicated than the rank of any one map.

There is a natural tensor product on representations of Q, which allows us to construct a representation ring R(Q) a la Grothendieck. We showed that the rank functors above commute with direct sum and tensor product of representations (addition and multiplication in R(Q)), and fix the identity, hence induce ring homomorphisms from R(Q) to the integers.  In more recent work, when Q is a tree quiver with a unique sink, we use combinatorial methods to construct all rank functors on Q and show that ring R(Q), modulo its ideal of nilpotents, is finitely generated as an abelian group.

Mark Kleiner, Preprojective elements of Coxeter groups.

Abstract: TBA

Mara Neusel, On the Hilbert Ideal.

Abstract: TBA

Charles Paquette, Triangulated stable module categories

Abstract: Happel proved that if an Artin algebra A is self-injective, then the stable module category is triangulated.  In this talk, we will look at the converse of this statement.  We will prove that the stable module category of A is triangulated if and only if A is stably equivalent to a self-injective algebra.  The latter is equivalent to A self-injective or Nakayama of Loewy length two (when A is connected).  In particular, we recover a result of I. Reiten about algebras stably equivalent to a self-injective algebra.

Franco Saliola, Two algebras related to finite Coxeter groups.

Abstract: This talk will focus on two closely-related algebras associated to finite Coxeter groups. The first is Solomon's descent algebra, a highly-structured subalgebra of the group algebra of the Coxeter group. The second algebra is a semigroup algebra constructed geometrically from a hyperplane arrangement associated to the Coxeter group. We will explore the connection between these algebras, applications to algebra, combinatorics and probability, and the relationship between the representation theories of these algebras. We will end by mentioning some open problems.

Ralf Schiffler, Cluster-tilted algebras and slices.

Abstract: Cluster-tilted algebras form a class of finite-dimensional algebras which provide a connection between cluster algebras and tilting theory, via cluster categories. In this talk, I will define cluster-tilted algebras and explain some of their properties, in particular the existence of local slices in their module category.

Hugh Thomas, Coloured quiver mutation for higher cluster categories.

Abstract: In a cluster category, if T is a (basic) cluster tilting object, then the quiver for End(T) is the crucial combinatorial datum associated to T.  One important property of this quiver is that if one knows the quiver for End(T), then one can determine the quiver for the endomorphism ring of a mutation of T, by applying Fomin-Zelevinsky mutation.

Simple examples show that when we move to higher cluster categories (that is to say, orbit categories of the form Cm=Db(H)/[m]τ-1), the endomorphism ring of an m-cluster-tilting object does not have this property; indeed, the endomorphism ring can be quite trivial.  We show how to define a quiver associated  to an
m-cluster-tilting object T in Cm whose edges are coloured with one of m+1 colours, which includes the quiver of End(T) as the 0-coloured arrows, and which has the property that, given the coloured quiver associated to T, one can determine the coloured quiver associated to any mutation of T, via a mutation procedure for coloured quivers which generalizes Fomin-Zelevinsky mutation.

This is joint work with Aslak Bakke Buan.

Shih-Wei Yang, Combinatorics of Coxeter elements and cluster algebras of finite type.

Abstract: In joint work with Andrei Zelevinsky, we give a unified geometric  realization of the cluster algebra of an arbitrary finite type with principal coefficients at any acyclic cluster (all necessary notions from the theory of cluster algebras will be explained). Our construction is based on the combinatorial study of orbits of fundamental weights under the action of the cyclic group generated by
a Coxeter element. As an application, we obtain a much simplified description of the Cambrian fans by N.Reading and D.Speyer. Interestingly, very similar combinatorics appeared in a different context in recent work by A.Kirillov, Jr. and J.Thind.
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